

%\magnification=1200
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\centerline{\bighead EXERCISES FOR MATHEMATICS 138A}
\bigskip\medskip
\centerline{\Head  WINTER 2004}
\bigskip\bigskip
The references denote sections of the text for the course:
\medskip
\itemitem{}M. P. do Carmo, {\it Differential 
geometry of Curves and Surfaces},
Prentice-Hall, Saddle River NJ, 1976, ISBN 0--132--12589--7.
\bigskip\bigskip\bigskip
\centerline{\Head I$\,$.\qquad Classical Differential Geometry of Curves}
\bigskip\bigskip
\centerline{\bf I.1$\;$:\quad Cross products}
\bigskip
\centerline{(do Carmo, \S~1--4)}
\medskip
\centerline{\it Additional exercise}
\medskip
{\bf 1. } \quad 
Verify that the cross product of vectors in ${\bf R}^3$ satisfies
the {\it Jacobi identity\/}:
$${\bf a}\times \left({\bf b}\times {\bf c}\right)~+~
{\bf b}\times \left({\bf c}\times {\bf a}\right)~+~
{\bf c}\times \left({\bf a}\times {\bf b}\right)~~=~~{\bf 0}~.$$
\bigskip\bigskip
\centerline{\bf I.2$\;$:\quad Parametrized curves}
\bigskip
\centerline{(do Carmo, \S~1--2)}
\medskip
do Carmo, \S~1--2, p. 5:~1, 2

do Carmo, \S~1--4, p. 15:~8
\smallskip
\itemitem{}[{\it Hints for p. $15${\#}$8$: } Strictly speaking there 
are two cases
depending upon whether the lines in question intersect.  Suppose
that they do not intersect.  In this case the shortest distance
between the lies is givey by a common perpendicular.  Assume
that the parametrizations are chosen so that {\bf u} and 
{\bf v} lie on this common perpendicular.  You may assume the
existence of a common perpendicular when working the problem.]
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Prove that a regular smooth curve lies on a 
straight line if and only if there is a point that lies on 
all its tangent lines.
\bigskip\bigskip
\centerline{\bf I.3$\;$:\quad Arc length and reparametrization}
\bigskip
\centerline{(do Carmo, \S~1--3)}
\medskip
do Carmo, \S~1--3, pp. 8--11:~7
\vfil\eject
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
(a) Given $a>0$, consider the set of all continuously differentiable
real valued functions $f$ on $[0,1]$ such that $f(0) = 0$ and
$f(1) = a$. Define $L(f)$ by the formula
$L(f)~~=~~\int_0^a\,\left|f'(t)\right|\,dt~.$
Show that the minimum value of $L(f)$ is $a$, and if 
equality holds then $f'$ is everywhere nonnegative.
[{\it Hints: } Since $f'\leq |f'|$ a similar inequality
holds for their definite integrals.  This inequality
of integrals is strict if and only if
$f'(t) <|f'(t)|$ for some $t$, which 
happens if and only if $f'(t)<0$ for that choice of $t$.]
\medskip
(b) Let $\rho$, $\theta$ and $\phi$ denote the usual spherical
coordinates, and suppose we have a curve on the sphere of radius
1 about the origin with parametric equations of the form
$${\bf x}(t)~~=~~\left(\cos\theta(t)\,\cos\sin(t),\, 
\sin\theta(t)\,\sin\phi(t),\,\cos\phi(t)\, \right)$$
for continuously differentiable functions $\theta(t)$ and
$\phi(t)$.  Prove that the length of this curve is given
by the formula
$$\int_a^b~\sqrt{(\theta'(t))^2~+~\sin^2\theta(t)\,({\phi\,}'(t))^2}\,
dt$$
where the curve is defined on $[a,b]$.
\medskip
(c) Show that among all regular smooth curves {\bf x} that are
defined on $[0,1]$, have images on the unit spere, and connect the 
points  $(1,0,0)$ and $(\cos a, \sin a, 0)$ for some  $a < \pi$,
the curve of shortest length
is given by the great circle arc joining the endpoints, 
and that any other curve with this length is a weak
reparametrization of the great circle arc ({\it i.e.}, if
$\alpha$ is the standard great circle arc, then any other
curve $\beta$ must have the form $\beta(t) = \alpha(\,f(t)\,)$,
where $f$ is a function from $[0,1]$ to itself that is continuously
differentiable and satisfies $f'\geq 0$.
[{\it Hints: } 
TO BE INSERTED. ]
%Let {\bf y} be the  curve in the $xy$-plane
%obtained from {\bf x} by replacing $\phi(t)$ with $\pi/2$.
%Why is the length of {\bf x} greater than or equal to
%the length of  {\bf y}?  And why is there strict inequality if
%${\phi\,}'(t_0)\,\sin \theta(t) \neq 0$ somewhere?  Why does this
%mean that the plane curve  $(cos \theta(t),\,\sin\theta(t),0)$
%is a weak reparametrization of $(\cos a\,t, \sin a\,t,0$?
%Recall that by continuity the
%latter implies ${\phi\,}'(t)\neq 0$ for all $t$ sufficiently close to
%$t_0$.  What does part (a) imply if $\phi$ is constant?] 
\medskip
{\sl Note. }  The final part of the problem
is a special case of the well known result that the shortest
curve on a sphere joining two points is given by the smaller of
the arcs on the great circle through the points; in fact,
one can use this special case to prove the general statement.
[A file containing a detailed proof will probably be inserted
into the course directory eventually.]
\bigskip\bigskip
\centerline{\bf I.4$\;$:\quad Curvature and torsion}
\bigskip
\centerline{(do Carmo, \S\S~1--5, 1--6)}
\medskip
do Carmo, \S~1--5, pp. 22-26:~11, 12$cd$

do Carmo, \S~1--6, pp. 29-30:~3
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Consider the problem of designing a set of railroad tracks that
contains a pair of parallel tracks along with a third going
from the first to the second smoothly.  Mathematically, the parallel
tracks themselves
may be viewed as corresponding to the parallel lines $y=0$ and $y=1$ 
in the coordinate plane, and the track going from one to the 
other may be viewed as a regular smooth curve that is the graph
of a twice differentiable function $f$ such that $f(x)$ is zero 
if $t\leq 0$, $f(x) = 1$ if $t\geq 1$, and on $[0,1]$ the function
$f$ is given by a polynomial $p(x)$.  The existence of a 
second derivative ensures that the slope of the tangent line would
be a continuous function of $x$, and in addition we want to assume 
that {\sl the curvature is also a continuous function of $x$}.
Finad a polynomial $p(x)$ of degree 5 such that all the required
conditions are fulfilled.  [{\it Hint: }
If we are given a graph curve with parametric equatitons 
$(t,\,y(t))$, then the curvature at parameter value $t$ is
given by the formula
$$k(t)~~=~~{{|y''|}\over{\left(1 + (y')^2\right)^{3/2}}}$$
and one step in the argument is to use this fact to compute
$p''(0)$ and $p''(1)$.  In fact, the conditions of the 
problem uniquely specify the values of $p$ and its first and
second derivatives at both 0 and 1.  Why does this mean the only
values to find are the coefficients of $x^3$, $x^4$ and $x^5$?]
\medskip
\noindent{\it Extra credit. }
Graph the function $f$ using calculator or computer graphics.
\bigskip\bigskip   
\centerline{\bf I.5$\;$:\quad Frenet-Serret Formulas}
\bigskip
\centerline{(do Carmo, \S\S~1--5, 1--6, 4--Appendix)}
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Let {\bf x} be a regular smooth curve with a continuous third
derivative, and let $\left({\bf T},\,{\bf N},\,{\bf B}\,\right)$
be its Frenet trihedron.` Prove that there is a vector ${\bf W}$
(the {\it Darboux vector}) such that ${\bf T\,}' = {\bf W}\times
{\bf T}$, ${\bf N\,}' = {\bf W}\times {\bf N}$, and
${\bf B\,}' = {\bf W}\times {\bf B}$.  What is the length of {\bf W}?
\medskip
{\bf 2. } \quad 
If {\bf x} is defined for $t>0$ by the formula
$${\bf x(t)}~~=~~\left(\,t,\,{{1+t}\over t},\,{{1-t^2}\over t}\right)$$
show that {\bf x} is planar.
\bigskip\bigskip   
\bigskip\bigskip\bigskip
\centerline{\Head II$\,$.\qquad Closed Curves as Boundaries}
\bigskip\bigskip   
\centerline{\bf II.1$\;$:\quad Regions, limits and continuity}
\bigskip
\centerline{(do Carmo, 2--Appendix A, 5--Appendix)}
\medskip
\centerline{\it Additional exercises}
\bigskip\bigskip   
\centerline{\bf II.2$\;$:\quad Smooth mappings}
\bigskip
\centerline{(do Carmo, 2--Appendix B)}
\medskip
\centerline{\it Additional exercises}
\bigskip\bigskip   
\centerline{\bf II.3$\;$:\quad Inverse and Implicit Function Theorems}
\bigskip
\centerline{(do Carmo, 2--Appendix A, 5--Appendix)}
\medskip
\centerline{\it Additional exercises}
\bigskip\bigskip   
\centerline{\bf II.4$\;$:\quad \bf Global properties of plane curves}
\bigskip
\centerline{(do Carmo, \S 1--7)}
\medskip
\centerline{\it Additional exercises}
\medskip
\bye
{\bf 1.$\iist$ } \quad 
In the integers ${\bf Z}$ let $p$ be a fixed prime.  For each integer 
$a > 0$ let 
$$U_a (n) = \{~x~|~x = n + kp^a , \ {{\rm some}} \ k \in {\bf Z} \}~.$$
Prove that the sets $U_a (n)$ 
form a basis for some topology on ${\bf Z}$.  [{\it Hint: }  Let
$\nu_p(n)$ denote the largest nonnegative integer $k$ such that
$p^k$ divides $n$ and show that 
$${\bf d}_p (a,b) = {1\over {p^{\nu_p(a-b)}}}$$
defines a metric on ${\bf Z}$].
\medskip
{\bf 2.$\ist$ } \quad
Let $A \subset X$ be closed and let 
$U \subset A$ be open in $A$.  Let $V$ be any open subset of $X$ with 
$U \subset V$.  Prove that $U \cup (V-A)$ is open in $X$.
\medskip
{\bf 3. } \quad
Let $E$ be a subset of the topological space $X$.  Prove that every
open subset $A\subset E$ is also open in $X$ if and only if
$E$ itself is open in $X$.
\vfil\eject
\bigskip\bigskip   
\centerline{\bf II.2$\;$:\quad Closed sets and limit points}
\bigskip
\centerline{(Munkres, \S~17)}
\medskip
Munkres, \S~17, pp. 100--102:~2, 8(a), 8(c)$\ist$, 19$\ist$, 20$\ist$
\medskip
Edwards, \S~I.7, pp. 48--49:~~7.6
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 0}. \quad 
Prove or give a counterexample to the following statement:  If $U$ and
$V$ are disjoint open subsets of a topological space $X$, then 
their closures are also disjoint.
\medskip
{\bf 1}. \quad 
Give an example to show that in a metric space the closure of an open 
$\varepsilon$ disk about a point is not necessarily the set of all 
points whose distance from the center is $\leq \varepsilon$.  
\bigskip
\noindent{\bf Definition. }  A subspace $D$ of a topological space
$X$ is {\it dense} if ${\overline D} = X$; equivalently, it is
dense if and only if for every nonempty open subset $U\subset X$
we have $U\cap D\neq \emptyset$.
\bigskip
{\bf 2}. \quad 
For which spaces is $X$ the only dense subset of itself?
\medskip
{\bf 3. } \quad
Let $U$ and $V$ be open dense subsets of $X$.  Prove that $U \cap V$ is 
dense in $X$.  
\medskip
{\bf 4. } \quad
A subspace $A$ of a topological space $X$ is said to be {\it locally 
closed} if for each $a \in A$ there is an open neighborhood $U$ of $a$ in 
$X$ such that $U \cap A$ is closed in $U$.  Prove that $A$ is locally 
closed if and only if $A$ is the intersection of an open subset and 
a closed subset.
\medskip
{\bf 5. } \quad 
(a) Suppose that $D$ is dense in $X$, and let $A \subset X$.  Give an 
example to show that $A \cap D$ is not necessarily dense in $A$.

(b) Suppose that $A \subset B \subset X$ and $A$ is dense in $B$.  Prove 
that $A$ is dense in $\overline{B}$. 
\medskip
{\bf 6. } \quad
Let $E$ be a subset of the topological space $X$.  Prove that every
closed subset $A\subset E$ is also closed in $X$ if and only if
$E$ itself is closed in $X$.
\medskip
{\bf 7. } \quad
Given a topological space $X$ and a subset $A\subset X$, explain why
the closure of the interior of $A$ does not necessarily
contain $A$.
\medskip
{\bf 8.$\ist$ } \quad
If $U$ is an open subset of $X$ and $B$ is an arbitrary subset
of $X$, prove that
$U\cap {\overline B}\subset {\overline {U\cap B}}$.
\medskip
{\bf 9.$\ist$ } \quad
If $X$ is a topological space, then 
the {\it Kuratowski closure axioms} are the following properties of
the operation $A\to {\bf CL}(A)$ sending $A\subset X$ to its closure
${\overline A}$:

\itemitem{{\bf (C1)}} \qquad$A\subset {\bf CL}(A)$ for all $A\subset X$.

\itemitem{{\bf (C2)}} \qquad${\bf CL}\left({\bf CL}(A)\,\right) = {\bf CL}(A)$ 

\itemitem{{\bf (C3)}} \qquad${\bf CL}(A\cup B) = {\bf CL}(A)\cup 
{\bf CL}(B)$ for all $A,\,B\subset X$.

\itemitem{{\bf (C4)}} \qquad${\bf CL}(\emptyset) = \emptyset$.

\noindent
Given an arbitrary set $Y$ and a operation ${\bf CL}$ assigning to 
each subset $B\subset Y$ another subset ${\bf CL}(B)\subset Y$
such that {\bf (C1) -- (C4)} all hold, prove that there is a unique
topology {\bf T} on $Y$ such that for all $B\subset Y$, the set
${\bf CL}(B)$ is the closure of $B$ with respect to {\bf T}.
\medskip
{\bf 10. }\quad
Suppose that $X$ is a space such that $\{p\}$ is closed for all
$x\in X$, andn let $A\subset X$.  Prove the following
statements: 

(a) ${\bf L}(A)$ is closed in $X$.

(b) For each point $b\in {\bf L}(A)$ and open
set $U$ containing $b$, the intersection $U\cap A$ 
is infinite.
\vfil\eject
\medskip
{\bf 11.$\ist$ }
Suppose that $X$ is a set and that {\bf I} is an operation
on subsets of $X$ such that the following hold:

$(i)~~$ ${\bf I}(X) = X$.

$(ii)~$ ${\bf I}(A)\subset A$ for all $A\subset X$.

$(iii)$ ${\bf I}\left({\bf I}(A)\right) = {\bf I}(A)$ for all 
$A\subset X$.

$(iv)~$ ${\bf I}(A\cap B) = {\bf I}(A)\cap {\bf I}(B)$.

\noindent
Prove that there is a unique topology {\bf T} on $X$ such
that $U\in {\bf T}$ if and only if ${\bf I}(A) = A$.
\medskip
{\bf 12.$\ist$ }
If $X$ is a topological space and $A\subset X$ then the
{\it exterior} of $X$, denoted by Ext$(X)$, is defined
to be $X-{\overline A}$.  Prove that this construction
has the following properties:

(a)~~  ${\rm Ext}(A\cup B) = {\rm Ext}(A)\cap
{\rm Ext}(B)$.

(b)  ~~${\rm Ext}(A)\cap A = \emptyset$.

(c)  ~~${\rm Ext}(\emptyset) = X$.

(d)  ~~${\rm Ext}(A) \subset 
{\rm Ext}\left({\rm Ext}\left({\bf L}(A)\,\right)\,\right)$.
%\vfil\eject
\medskip
{\bf 13. } \quad
Let $A_1$ and $A_2$ be subsets of a topological space $X$, and
let $B$ be a subset of $A_1\cap A_2$ that is closed in both 
$A_1$ and $A_2$ with respect to the subspace to topologies
on each of these sets.  Prove that $B$ is closed in $A_1\cup A_2$.
\medskip
{\bf 14.$\ist$ } \quad
Suppose that $A$ is a closed subset of a topological space 
$X$ and $B$ is the closure of ${\rm Int}(A)$.  Prove that 
${\overline B}\subset A$ and ${\rm Int}(B) = {\rm Int}(A)$.
\medskip
{\bf 15. } \quad  Let $X$ be a topological space and let
$A\subset Y\subset X$.

(a) Prove that the interior of $A$ with respect to $X$ is contained
in the interior of $A$ with respect to $Y$.

(b)$\ist$ Prove that the boundary of $A$ with respect to $Y$ is contained
in the intersection of $Y$ with the boundary of $A$ with respect to $X$.

(c)$\ist$ Give examples to show that the inclusions in the preceding two
statements may be proper (it suffices to give one example for which
both inclusions are proper).
\bigskip\bigskip   
\centerline{\bf II.3$\;$:\quad Continuous functions}
\bigskip
\centerline{(Munkres, \S\S~18, 21; Edwards, \S~I.7)}
\medskip
Munkres, \S~18, pp. 111--112:~2, 6$\ist$, 8(a) with $Y = {\bf R}$, 9(c)
\medskip
Edwards, \S~I.7, p. 48:~~7.3, 7.4
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Give examples of continuous functions from {\bf R} to itself that
are neither open nor closed.
\medskip
{\bf 2. } \quad Let $X$ be a topological space, and let
$f,\,g:X\to {\bf R}$ be continuous.  Prove that the functions
$|f|$, max$(f,g)$ [whose value at $x\in X$ is the larger
of $f(x)$ and $g(x)$] and min$(f,g)$ [whose value at $x\in X$ 
is the smaller of $f(x)$ and $g(x)$] are all continuous.
[{\it Hints: }  If $h:X\to {\bf R}$ is continuous, what can
one say about the sets of points where $h=0$, $h<0$ and
$h>0$?  What happens if we take $h=f-g$?]
\medskip
{\bf 3.$\ist$ } \quad Let $f:X\to Y$ be a set-theoretic mapping
of topological spaces.  

(a) Prove that $f$ is open if and only
if $f(\,{\rm Int}(A)\,)\subset {\rm Int} (f(A))$ for all $A\subset X$
and that $f$ is closed if and only if ${\overline {f(A)}}
\subset f({\overline A})$ for all $A\subset X$.  

(b) Using this
and other results from the course notes, prove that $f$ is
continuous and closed if and only if ${\overline {f(A)}} = 
f({\overline A})$ for all $A\subset X$ and $f$ is 
is continuous and open if and only if $f^{-1}(\,{\rm Int}(B)\,)
\subset {\rm Int}(f(B))$ for all $B\subset Y$.
\bigskip\bigskip   
\vfil\eject
\centerline{\bf II.4$\;$:\quad Cartesian products}
\bigskip
\centerline{(Munkres, \S\S~15, 19)}
\medskip
Munkres, \S~18, pp. 111--112:~10, 11
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1}.$\iist$ \quad 
(``A product of products is a product,'')\qquad
Let $\{ A_\alpha~|~\alpha \in 
{\cal A} \}$ be a family of nonempty sets, and let ${\cal A} = 
\cup \{ {\cal A}_\beta~|~\beta \in {\cal B} \}$ be a partition of 
${\cal A}$.  Construct a bijective map of $\prod \{ A_\alpha~|~\alpha \in 
{\cal A} \}$ to the set 
$$\prod_{\beta} \bigl\{ {\textstyle{\prod}} \{ A_\alpha~|~\alpha \in 
{\cal A}_{\beta} \} \ \bigr\}~ .$$
If each $A_\alpha$ is a topological space and we are working with product
topologies, prove that this bijection is a homeomorphism.
\medskip
{\bf 2.$\iist$ } \quad
{\it Non-Hausdorff topology.}  In topology courses one is ultimately 
interested in spaces that are Hausdorff.  However, there are 
contexts in which certain types of non-Hausdorff spaces arise (for 
example, the Zariski topologies in algebraic geometry, which are 
defined in most textbooks on that subject).

(a) A topological space $X$ is said to be {\it irreducible} if it 
cannot be written as a union $X = A \cup B$, where $A$ and $B$ are 
proper closed subspaces of $X$.  Show that $X$ is irreducible if and only 
if every pair of nonempty open subsets has a nonempty intersection.  Using 
this, show that an open subset of an irreducible space is irreducible.  

(b) Show that every set with the indiscrete topology is irreducible, 
and every infinite set with the finite complement topology is irreducible.  

(c) Show that an irreducible Hausdorff space contains at most one point.  
\medskip
{\bf 3}. \quad 
Let $f : X \rightarrow Y$ be a map, and define the {\it graph} of $f$ to 
be the set $\Gamma_f$ of all points $(x , y) \in X \times Y$ such that 
$y = f(x)$.  Prove that the map $x \rightarrow (x , f(x) )$ is a 
homeomorphism from $X$ to $\Gamma_f$ if and only if $f$ is continuous.  
\medskip
{\bf 4. } \quad
Let $X$ be a topological space that is a union of two closed
subspaces $A$ and $B$, where each of $A$ and $B$ is Hausdorff
in the subspace topology.  Prove that $X$ is Hausdorff.
\medskip
{\bf 5.$\ist$ } \quad  
Let $A$ be some nonempty set, let
$\{ X_\alpha~|~\alpha \in {\cal A} \}$ 
and $\{ Y_\alpha~|~\alpha \in {\cal A} \}$ 
be families of topological spaces, and for each 
$\alpha\in A$ suppose that $f_\alpha:X_\alpha\to
Y_\alpha$ is a homeomorphism.  Prove that the product map
$$\prod_{\alpha} f_\alpha\;:~~\prod_\alpha X_\alpha\longrightarrow
\prod_\alpha Y_\alpha$$
is also a homeomorphism.  [{\it Hint: } What happens when you
take the product of the inverse maps?]
\medskip
{\bf 6. } \quad  
Let $X$ be a topological space and let $T:X\times X\times X\to
X\times X\times X$ be the map that cyclically permutes the
coordinates:  $T(x,y,z) = (z,x,y)$  Prove that $T$ is a 
homeomorphism.  [{\it Hint: }  What is the test for continuity
of a map into a product?  Can you write down an explicit
formula for the inverse function?]
\bigskip
\bigskip\bigskip   
\bigskip\bigskip\bigskip
\vfil\eject
\centerline{\Head III$\,$.\qquad Spaces with special properties}
\bigskip\bigskip
\centerline{\bf III.1$\;$:\quad Compact spaces -- I}
\bigskip
\centerline{(Munkres, \S\S~26, 27)}
\medskip
Munkres, \S~26, pp. 170--172:~3, 7$\ist$, 8$\ist$

Munkres, \S~27, pp. 177--178:~2(a) (b) (c) (d) (e)$\ist$
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Let $X$ be a compact Hausdorff space, and let $f: X \rightarrow X$ be
continuous.  Define $X_1 = X$ and $X_{n+1} = f ( X_n )$, and set $A =
\bigcap_n X_n$.  Prove that $A$ is a nonempty subset of $X$ and $f(A) =
A$.  
\medskip
{\bf 2.$\iist$ } \quad 
A topological space $X$ is said to be a $k$-{\bf space} if
it satisfies the following condition:  {\it A subset $A\subset X$
is closed if and only if for all compact subsets $K\subset X$,
the intersection $A\cap K$ is closed. }  It turns out that a large
number of the topological spaces one encounters in topology,
geometry and analysis are $k$-spaces (including all metric spaces
and compact Hausdorff spaces), and the textbooks by Kelley and 
Dugundji contain a great deal of information about these $k$-spaces 
(another important reference is the following paper by N. E.
Steenrod:  {\it A convenient category of topological spaces}, 
Michigan Mathematical Journal {\bf 14} (1967), pp. 133--152).

(a) Prove that if $(X,{\bf T})$ is a Hausdorff topological space 
then there is a unique minimal topology ${\bf T}^\kappa$ containing
${\bf T}$ such that 
${\cal K}(X) = (X,{\bf T}^\kappa)$ is a Hausdorff $k$-space.

(b) Prove that if $f:X\to Y$ is a continuous map of 
Hausdorff topological spaces, then $f$ is also continuous when
viewed as a map from ${\cal K}(X)\to {\cal K}(Y)$.
\medskip
{\bf 3.$\iist$ } \quad
{\it Non-Hausdorff topology revisited.}  A topological space $X$ 
is {\it noetherian} if every non\-empty family of open subsets has a maximal 
element.  This class of spaces is also of interest in algebraic
geometry.

(a) ({\it Ascending Chain Condition}) Show that a space $X$ is noetherian 
if and only if every increasing sequence of open subsets 
$$ U_1~ \subset~ U_2~ \subset ~...~ $$ 
stabilizes; {\it i.e.}, there is some positive integer $N$ such that 
$n \geq N$ implies $U_n = U_N$.  

(b) Show that a space $X$ is noetherian if and only if every 
open subset is compact.  

(c) Show that a noetherian Hausdorff space is finite (with the discrete 
topology).  [{\it Hint: } Show that every open subset is closed.]  

(d) Show that a subspace of a noetherian space is noetherian.
\bigskip\bigskip
\centerline{\bf III.2$\;$:\quad Complete metric spaces}
\bigskip
\centerline{(Munkres, \S\S~43, 45)}
\medskip
Munkres, \S~43, pp. 270--271:~1, 3(b), 6(a)(c)$\ist$
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Show that the Nested Intersection Property  for complete
metric spaces  does not necessarily hold for nested sequences of
closed subsets $\{\,A_n\,\}$ if $\lim_{n\to\infty} {\rm diam}
(A_n) \neq 0$; {\it i.e.}, in such cases one might have
$\cap_n~A_n = \emptyset$. [{\it Hint: }  Consider the set $A_n$
of all continuous functions $f$ from $[0,1]$ to itself that
are zero on $[{1\over n},1]$ and also satisfy $f(0)=1$.]
\medskip
{\bf 2.$\iist$ }\quad 
Let $\ell^2$ be the set of all real sequences ${\bf x} =  \{\,x_n\,\}$ 
such that $\sum_n |x_n|^2$ converges.  The results of Exercise 10 on 
page 128 of Munkres show that $\ell^2$ is a normed vector space with 
the norm
$$\bigl|\,{\bf x}\,\bigr| = \left(\sum_n |x_n|^2\right)^{1/2}~.$$
Prove that $\ell^2$ is complete with respect to the
associated  metric.  [{\it Hint: } If ${\bf p}_i$ gives the
$i^{{\rm th}}$ term of an element in $\ell^2$, show that ${\bf p}_i$
takes Cauchy sequences to Cauchy sequences.  This gives a candidate
for the limit of a Cauchy sequence in $\ell^2$.  Show that this
limit candidate actually lies in $\ell^2$ and that the Cauchy
sequence converges to it. See also Royden, {\it Real Analysis}, Section
6.3, pages 123--127, and also
Rudin, {\it Principles of Mathematical Analysis}, Theorem
11.42 on page 329--330 together with 
the discussion on the following two pages.]
\bigskip\bigskip
\centerline{\bf III.3$\;$:\quad Implications of completeness}
\bigskip
\centerline{(Munkres, \S~48; Edwards, \S~III.1)}
\medskip
Munkres, \S~27, pp. 177--178:~6$\ist$

Munkres, \S~48, pp. 298--300:~1, 2$\ist$, 4$\ist$
\medskip
Edwards, \S~III.1, p. 171:~~1.6, 1.7$\ist$
\medskip
%\vfil\eject
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Let $A$ and $B$ be subspaces of $X$ and $Y$ respectively such
that $A\times B$ is nowhere dense in $X\times Y$ (with respect
to the product topology).  Prove that either $A$ is nowhere dense
in $X$ or $B$ is nowhere dense in $Y$, and give an example to
show that ``or'' cannot be replaced by ``and.''
\medskip
{\bf 2. } \quad 
Is there an uncountable topological space of the first category?
\medskip
{\bf 3.$\ist$ } \quad 
Let $X$ be a metric space.  A map $f:X\to X$ is said to be an {\it 
expanding similarity} of $X$ if $f$ is onto and there is a constant
$C>1$ such that 
$${\bf d}\left(\,f(u),\,f(v)\,\right) ~=~~ C\cdot{\bf d}(u,v)$$
for all $u,\,v\in X$ (hence $f$ is 1--1 and uniformly continuous).
Prove that every expanding similarity of a complete metric space has a
unique fixed point.  [{\it Hint and comment: } Why does $f$ have an 
inverse that 
is uniformly continuous, and why does $f(x) = x$ hold if and only
if $f^{-1}(x) = x$? If $X= {\bf R}^n$ and a similarity is given
by $f(x) = cAx+b$ where $A$ comes from an orthogonal matrix and
either $0<c<1$ or $c>1$, then one can prove the existence of a 
unique fixed point directly using linear algebra.]
\bigskip\bigskip
\vfil\eject
\centerline{\bf III.4$\;$:\quad Connected spaces}
\bigskip
\centerline{(Munkres, \S\S~23, 24, 25)}
\medskip
Munkres, \S~23, p. 152:~3, 4, 5, 9, 12$\ist$ assuming $Y$ is closed

Munkres, \S~24, pp. 157--159:~1(b)
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Prove that a topological space $X$ is connected if and only if every 
open covering ${\cal U} = \{ U_\alpha \}$ has the following property:  For 
each pair of sets $U,\ V$ in ${\cal U}$ there is a sequence of
open sets $\{ U_0, U_1, ... U_n \}$ in ${\cal U}$ such that $U = U_0$, 
$V = U_n$, and $U_i \cap U_{i+1} \neq \emptyset$ for all $i$.  
\medskip
{\bf 2. } \quad 
Let $X$ be a connected space, and let ${\cal R}$ be an equivalence relation 
that is locally constant (for each point $x$ all points in some 
neighborhood of $x$ lie in the ${\cal R}$-equivalence class of $x$).  Prove 
that $\cal R$ has exactly one equivalence class.  
\medskip
{\bf 3. } \quad 
Prove that an open subset in ${\bf R}^n$ can have at most 
countably many components.  Give an example to show this is not 
necessarily true for closed sets.  
%\vfil\eject
\bigskip\bigskip
\centerline{\bf III.5$\;$:\quad Variants of connectedness}
\bigskip
\centerline{(Munkres, \S\S~23, 24, 25)}
\medskip
Munkres, \S~24, pp. 157--159:~8(b)

Munkres, \S~25, pp. 162--163:~10(a) (b) (c)$\ist$ only Examples $A$ and $B$
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Prove that a compact locally connected space has only finitely many
components.
\medskip
{\bf 2.$\ist$ } \quad 
Give and example to show that if $X$ and $Y$ are locally
connected metric spaces and $f:X\to Y$ is continuous then
$f(X)$ is not necessarily locally connected.
%\vfil\eject
\bigskip\bigskip\bigskip
\centerline{\Head IV$\,$.\qquad Smooth mappings}
\bigskip\bigskip
\centerline{\bf IV.1$\;$:\quad Linear approximations}
\bigskip
\centerline{(Edwards, \S\S~II.1, II.2)}
\medskip
Edwards, \S\S~II.1, p. 75:~~2.6$\ist$, 2.8$\ist$

Edwards, \S\S~II.2, pp. 88--90:~~3.3$\ist$, 3.16$\ist$
\bigskip\bigskip
\centerline{\bf IV.2$\;$:\quad Properties of smooth functions}
\bigskip
\centerline{(Edwards, \S~II.3)}
\medskip
Edwards, \S~II.3, pp. 88--90:~~3.4$\ist$, 3.11, 3.12$\ist$, 3.13$\ist$, 3.14

Edwards, \S\S~III.2, p. 180:~~2.1$\ist$, 2.8$\iist$
\vfil\eject
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad  
(a) Suppose that $X$ and $Y$ are subsets of
${\bf R}^n$ and ${\bf R}^m$ respectively and that
$f:X\to Y$ and $g:Y\to {\bf R}^p$ are maps that 
satisfy Lipschitz conditions.  Prove that the
composite $g\tinycirc f$ also satisfies a Lipschitz
condition.  

(b) Suppose that $X\subset {\bf R}^n$, and let
$f,\,g:X\to {\bf R}^m$ and $h:X\to {\bf R}$ satisfy 
Lipschitz conditions.  Prove that $f+g$ satisfies a 
Lipschitz condition and if $X$ is compact then 
$h\cdot f$ also satisfies a Lipschitz condition.  
If $h>0$ and $X$ is compact, does $1/h$ satisfy a 
Lipschitz condition?  
Prove this or give a counterexample.  

(c) Suppose that $X\subset {\bf R}^n$, and let
$f:X\to {\bf R}^m$ be given.  Prove that $f$
satisfies a Lipschitz condition if and only if
all of its coordinate functions do.
\medskip
{\bf 2.$\ist$ } \quad  
In the notation of the preceding exercise, 
suppose that $X=A\cup B$ and that $f$ is 
continuous and satisfies Lipschitz conditions 
on $A$ and $B$ as
well as on an open neighborhood of $A\cap B$.  Does
$f$ satisfy a Lipschitz condition on $A\cup B$?
Prove this or give a counterexample.  What
happens if we assume $A$ and $B$ are compact?
Justify your answer.
%\vfil\eject
\bigskip\bigskip
\centerline{\bf IV.3$\;$:\quad Inverse Function Theorem}
\bigskip
\centerline{(Edwards, \S\S~III.2, III.3)}
\medskip
Edwards, \S\S~III.3, pp. 194--196:~~3.1, 3.3$\ist$, 3.4$\ist$, 3.14$\ist$
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 0. } \quad
Prove that $F(x,y) = (e^x+y,\,x-y)$ defines a 
${\bf C}^\infty$ homeomorphism of ${\bf R}^2$ with a 
${\bf C}^\infty$ inverse.
\bigskip
{\bf 1. } \quad
Prove that $F(x,y) = (xe^y+y,\,xe^y-y)$ defines a 
${\bf C}^\infty$ homeomorphism of ${\bf R}^2$ with a 
${\bf C}^\infty$ inverse.
\bigskip
{\bf 2.$\ist$ }
Prove that 
$$F(x,y,z) = \left({{x}\over{2+y^2}} + ye^z, 
{{x}\over{2+y^2}} - ye^z,  2ye^z + z\right)$$
defines a 
${\bf C}^\infty$ homeomorphism of ${\bf R}^3$ with a 
${\bf C}^\infty$ inverse.
\bigskip
{\bf 3. }
Let $f(x,y) = (x+y, x^2+y)$.  Check that $f$  meets the conditions
to have a local inverse near $f(1,0) = (1,1)$, and if $g$ is this
local inverse find $Dg(1,1)$ without finding a formula for the inverse
function explicitly.
\bigskip
{\bf 4.$\ist$ }
Consider the mapping $f:{\bf R}^2\to {\bf R}^2$ given by $f(x,y) = 
(x^2+y^2, 2xy)$.  Show that the Jacobian vanishes on the lines 
$y=\pm\, x$.  What is the image of $f$?  [{\it Hint: } Try using polar 
coordinates.]  The Inverse Function Theorem guarantees that $f$ has
a local inverse at $f(1,0) = (1,0)$.  Find the inverse explicitly and
describe a region on which it is defined.
\bigskip
{\bf 5. }
Let $f:{\bf R}^2\to{\bf R}^2$ be defined by $f(v,w) = 
(v+w+e^{vw},w+2v+e^{-vw})$.  Prove that there is a local 
inverse at $f(0,0) = (1,1)$ and if 
$$g(x,y) = \left(\,v(x,y)\,,~w(x,y)\,\right)$$
show that
$${{\partial v}\over{\partial x}}(1,1) + 
{{\partial w}\over{\partial x}}(1,1) = 
{{\partial v}\over{\partial y}}(1,1)~.$$
%\vfil\eject
\bigskip
\bigskip\bigskip
\bigskip\bigskip\bigskip
\centerline{\Head V$\,$.\qquad Constructions on spaces}
\bigskip\bigskip
\centerline{\bf V.1$\;$:\quad Quotient spaces}
\bigskip
\centerline{(Munkres, \S~22)}
\medskip
Munkres, \S~22, pp. 144--145:~4
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 0}. \quad 
Suppose that $X$ is a space with the discrete topology and $\cal R$
is an equivalence relation on $X$.  Prove that the quotient
topology on $X/{\cal R}$ is discrete.
\medskip
{\bf 1}.$\ist$ \quad 
If $A$ is a subspace of $X$, a continuous map $r: X \rightarrow A$ 
is called a {\it retraction} if the restriction of $r$ to $A$ is the 
identity.  Show that a retraction is a quotient map.  
\medskip
{\bf 2}. \quad 
Let $\cal R$ be an equivalence relation 
on a space $X$, and assume that $A \subset X$ contains points from 
every equivalence class of $\cal R$.  Let ${\cal R}_0$ be the induced 
equivalence relation on $A$, and let 
$$j: A / {\cal R}_0 \rightarrow X / {\cal R}$$
be the associated $1-1$ correspondence of equivalence classes.  Prove that 
$j$ is a homeomorphism if there is a retraction $r: X \rightarrow A$ such 
that each set $r^{-1} ( \{ a \} )$ is contained in an ${\cal R}$-equivalence 
\medskip
{\bf 3}. \quad 
(a) Let 0 denote the origin in ${\bf R}^3$.  In ${\bf R}^3 - \{ 0 \}$ 
define $x{\cal R}y$ if $y$ is a nonzero multiple of $x$ (geometrically, if 
$x$ and $y$ lie on a line through the origin).  Show that ${\cal R}$ is an 
equivalence relation; the quotient space is called the {\it real 
projective plane} and denoted by 
${\bf R}{\bf P}^2$.  

(b) Using the previous exercise show that ${\bf RP}^2$ can also be 
viewed as the quotient of $S^2$ modulo the equivalence relation
$x\sim y \Longleftrightarrow y = \pm\, x$.  
In particular, this shows that ${\bf RP}^2$ is compact.
[{\it Hint: } Let $r$ be the radial compression map 
that sends $v$ to $|v|^{-1} v$.]
\medskip
{\bf 4.$\iist$ } \quad
In $D^2 = \{ x \in {\bf R}^2 | \ |x| \leq 1 \}$, consider the 
equivalence relation generated by the condition $x{{\cal R}'}y$ if 
$|x| = |y| = 1$ and $y = -x$.  Show that this quotient space is 
homeomorphic to ${\bf R}{\bf P}^2$.  
\medskip
[{\it Hints:} Use the description of ${\bf RP}^2$ as a quotient
space of $S^2$ from the previous exercise, and let 
$h: D^2 \rightarrow S^2$ be defined by 
$$h (x,y) = (x,y, \sqrt{\, 1 - x^2 - y^2\;} ).$$
Verify that $h$ preserves equivalence classes and therefore induces a 
continuous map $\overline{h}$ on quotient spaces.  
Why is $\overline{h}$ a $1-1$ and onto mapping?  
Finally, prove that ${\bf RP}^2$ is Hausdorff and 
$\overline{h}$ is a closed mapping.] 
\medskip
{\bf 5. } \quad
Suppose that $X$ is a topological space with topology {\bf T},
and suppose also that $Y$ and $Z$ are sets with set-theoretic maps
$f:X\to Y$ and $g:Y\to Z$.  Prove that the quotient topologies 
satisfy the condition
$$(g\tinycirc f)_*{\bf T} = g_*\left(f_*{\bf T}\right)~.$$
\noindent
(Informally, a quotient of a quotient is a quotient.)
\medskip
{\bf 6.$\ist$ } \quad
If $Y$ is a topological space with a topology {\bf T} and
$f;X\to Y$ is a set-theoretic map, then the {\it induced 
topology} $f^*{\bf T}$ on $X$ is defined to be the set of
all subsets $W\subset X$ having the form $f^{-1}(U)$
for some open set $U\in {\bf T}$.  Prove that $f^*{\bf T}$
defines a topology on $X$, that it is the unique smallest 
topology on $X$ for which $f$ is continuous, and that if
$h:Z\to X$ is another set-theoretic map then
$$(f\tinycirc h)^*{\bf T} = h^*\left(f^*{\bf T}\right)~.$$
\medskip
{\bf 7. } \quad 
Let $X$ and $Y$ be topological spaces, and define an
equivalence relation $\cal R$ on $X\times Y$ by
$(x,y)\sim (x',y')$ if and only if $x = x'$.  Show
that $X\times Y/{\cal R}$ is homeomorphic to $X$.
\medskip
{\bf 8. } \quad 
Let ${\cal R}$ be an equivalence relation on a topological 
space $X$, let $\Gamma_{\cal R}$ be the graph of $\cal R$,
and let $\pi:X\to X/{\cal R}$ be the quotient projection.
Prove the following statements:

(a) If $X/{\cal R}$ satisfies the Hausdorff Separation Property
then $\Gamma_{\cal R}$ is closed in $X\times X$.

(b) If $\Gamma_{\cal R}$ is closed and $\pi$ is open, then
$X/{\cal R}$ is Hausdorff.

(c) If $\Gamma_{\cal R}$ is open then $X/{\cal R}$ is
discrete.
\bigskip\bigskip
\centerline{\bf V.2$\;$:\quad Sums and cutting and pasting}
\bigskip
\centerline{({not covered in the texts})}
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
Let $\{ A_\alpha~|~\alpha \in 
{\cal A} \}$ be a family of topological spaces, and let
$X = \coprod_\alpha A_\alpha$.  Prove that $X$ is locally 
connected if and only if each $A_\alpha$ is locally connected.
\medskip
{\bf 2. } \quad 
In the preceding exercise, formulate and prove necessary and
sufficient conditions on $\cal A$ and the sets $A_\alpha$ for
the space $X$ to be compact.
\medskip
{\bf 3.$\iist$ } \quad 
Prove that ${\bf RP}^2$ can be constructed by identifying the
edge of a M\"obius strip with the edge circle on a closed 
2-dimensional disk by filling in the details of the 
following argument:  Let $A\subset S^2$ be
the set of all points $(x, y, z)\in S^2$ such that
$|z|\leq{1\over 2}$, and let $B$ be the set of all points where
$|z|\geq{1\over 2}$.  If $T(x) = -x$, then $T(A) = A$ and
$T(B)=B$ so that each of $A$ and $B$ (as well as their
intersection) can be viewed as a union of equivalence classes
for the equivalence relation that produces ${\bf RP}^2$.
By construction $B$ is a disjoint union of two pieces 
$B_\pm$ consisting of all points where sign$(z) = \pm\, 1$,
and thus it follows that the image of $B$ in the quotient 
space is homeomorphic to $B_+\cong D^2$.  Now consider
$A$.  There is a homeomorphism $h$ from $S^1\times[-1,1]$ to 
$A$ sending $(x,y,t)$ to $(\alpha(t) x, \alpha(t) y, 
{1\over 2}t)$  where 
$$\alpha(t) = \sqrt{1-{{t^2}\over 4}}$$
and by construction $h(-v) = -h(v)$.  The image of $A$ in
the quotient space is thus the quotient of $S^1\times
[-1,1]$ modulo the equivalence relation $u\sim v
\Longleftrightarrow u = \pm v$.  This quotient space is
in turn homeomorphic to the quotient space of the
upper semicircular arc $S^1_+$ (all points with nonnegative
$y$-coordinate) modulo the equivalence relation 
generated by setting $(-1,0,t)$ equivalent to 
$(1,0,-t)$, which yields the M\"obius strip.  The 
intersection of this subset in the quotient with the
image of $B$ is just the image of the closed curve on the
edge of $B_+$, which also represents the edge curve on
the M\"obius strip.
\medskip
{\bf 4. } \quad 
Suppose that the topological space $X$ is a union of two
closed subspaces $A$ and $B$, let $C = A\cap B$, let
$h:C\to C$ be a homeomorphism, and let $A\cup_h B$ be
the space formed from  $A \sqcup B$ by indentifying 
$x\in C\subset A$ with $h(x)\in C\subset B$.  Prove that
$A\cup_h B$ is homeomorphic to $X$ if $h$ extenhds to a 
homeomorphism $H:A\to A$, and give an example for which
$X$ is not homeomorphic to $A\cup_h B$.  [{\it Hint: }
Construct the homeomorphism using $H$ in the first case,
and consider also the case where $X = S^1 \sqcup S^1$,
with $A_\pm = = S^1_\pm \sqcup S^1_\pm$; then $C = 
\{\pm\,1\}\times\{1,2\}$, and there is a homeomorphism
from $h$ to itself such that $A_{+}\cup_h A_{-}$ is
connected.]
\medskip
{\bf 5.$\iist$ } \quad 
{\it One-point unions. }
One conceptual problem with  the disjoint union of topological 
spaces is that it is never connected except for the trivial
case of one summand.  In many geometrical and topological 
contexts it is extremely useful to construct a modified version
of disjoint unions that is connected if all the pieces are.  Usually 
some additional structure is needed in order to make such constructions.  

In this exercise we shall describe such a construction for objects
known as {\it pointed spaces} that are indispensable for many
purposes ({\it e.g.}, the definition of fundamental groups as
in Munkres).  A pointed space is a pair $(X,x)$ consisting of 
a topological space $X$ and a point $x\in X$; we often call $x$ the
{\it base point}, and unless stated otherwise {\it the one point
set consisting of the base point is assumed to be closed}.  If 
$(Y,y)$ is another pointed space and 
$f:X\to Y$ is continuous, we shall say that $f$ {\it is a base point
preserving continuous map from $(X,x)$ to $(Y,y)$} if $f(x) = y$,
In this case we shall often write $f:(X,x)\to (Y,y)$.  Identity
maps are base point preserving, and composites of base point 
preserving maps are also base point preserving.

(a) Given a finite collection of pointed spaces $(X_i,x_i)$,
define an equivalence relation on $\coprod_i X_i$ whose 
equivalence classes consist of $\coprod_j \{x_j\}$ and all
one point sets $y$ such that  $y\not\in \coprod_j \{x_j\}$.
Define the {\it one point union} or {\it wedge}
$$\bigvee_{i=1}^n (X_j,x_j) = (X_1,x_1)\vee ~\cdots ~\vee
(X_n,x_n)$$
to be the quotient space of this equivalence relation with
the quotient topology.  The base point of this space is taken to be 
the  class of $\coprod_j \{x_j\}$.

(a) Prove that the wedge is a union of 
closed subspaces $Y_j$ such that each $Y_j$ is homeomorphic
to $X_j$ and if $j\neq k$ then $Y_j\cap Y_k$ is the base
point.  Explain why $\vee_k~(X_k,x_k)$ is Hausdorff if and
only if each $X_j$ is Hausdorff, why $\vee_k~(X_k,x_k)$ 
is compact if and only if each $X_j$ is compact, 
and why $\vee_k~(X_k,x_k)$ 
is connected if and only if each $X_j$ is connected
(and the same holds for arcwise connectedness).

(b) Let $\varphi_j:(X_j,x_j)\to \vee_k~(X_k,x_k)$ be the composite
of the injection $X_j\to \coprod_k~X_k$ with the quotient
projection; by construction $\varphi_j$ is base point preserving.
Suppose that $(Y,y)$ is some arbitrary pointed space and we are 
given a sequence of base point preserving continuous maps 
$F_j:(X_j,x_j)\to (Y,y)$.  Prove that there is a unique
base point preserving continuous mapping
$$F: \vee_k~(X_k,x_k)\to (Y,y)$$
such that $F\tinycirc \varphi_j = F_j$ for all $j$.

(c) In the infinite case one can carry out the set-theoretic
construction as above but some care is needed in defining the
topology.  Show that if each $X_j$ is Hausdorff and one takes 
the so-called {\it weak
topology} whose closed subsets are generated by the 
family of subsets $\varphi_j(F)$ where  $F$ is closed in $X_j$
for some $j$,
then $[1]$ a function  $h$  from the wedge into some other
space $Y$ is continuous if and only if each composite
$h\tinycirc \varphi_j$ is continuous, $[2]$ the existence 
and uniqueness theorem for mappings from the wedge
(in the previous portion of the exercise) generalizes to
infinite wedges with the so-called weak topologies.

(d) Suppose that we are given an infinite wedge such that
each summand is Hausdorff and contains at least two points.
Prove that the wedge with the so-called weak topology is
not compact.
\medskip
{\it Remark. }
If each of the summands in (d) is compact Hausdorff, then
there is a natural candidate for a {\it strong topology} on a 
countably infinite wedge which makes the latter into a compact 
Hausdorff space.  In some cases this topology can be viewed 
more geometrically; for example, if each $(X_j,x_j)$ is 
equal to $(S^1,1)$ and there are countably infinitely many 
of them, then the space one obtains is the {\it Hawaiian 
earring} in ${\bf R}^2$ given by the union of the circles 
defined by the equations 
$$\left(x-{1\over {2^k}}\right)^2 + y^2 = {1\over {2^{2k}}}~.$$
As usual, drawing a picture may be helpful.  The $k^{{\rm th}}$
circle has center $(1/2^k,\,0)$ and passes through the origin;
the $y$-axis is the tangent line to each circle at the origin.
%\vfil\eject
\bigskip\bigskip
\bigskip\bigskip\bigskip
\centerline{\Head VI$\,$.\qquad Spaces with additional properties}
\bigskip\bigskip
\centerline{\bf VI.1$\;$:\quad Second countable spaces}
\bigskip
\centerline{(Munkres, \S~30)}
\medskip
Munkres, \S~30, pp. 194--195: ~9 (first part only), 10, 13$\ist$, 14$\ist$
\medskip
%\vfil\eject
\centerline{\it Additional exercises}
\medskip
{\bf 1}. \quad 
If $(X,{\bf T})$ is a second countable Hausdorff space, prove that
the cardinalities of both $X$ and {\bf T} are less than or equal to
$2^{\aleph_0}$.  (Using the formulas for cardinal numbers in 
Section I.3 of the course notes and the separability of $X$ one
can prove a similar inequality for ${\bf BC}(X)$.)
\medskip
{\bf 2}.$\iist$ \quad 
{\it Separability and subspaces. }  The following example shows that
a closed subspace of a separable Hausdorff space is not necessarily
separable.

(a) Let $X$ be the upper half plane ${\bf R}\times [0,\infty)$ and
take the topology generated by the usual metric topology plus the 
following sets:
$$T_\varepsilon(x) = \{\,(x,0)\,\}~~\cup~~N_\varepsilon
\left(\,(x,\varepsilon)\,\right), ~~
{\rm where}~~x\in {\bf R}~~{\rm and} ~~\varepsilon~>~0$$
Geometrically, one takes the interior region of the circle in the
upper half plane that is tangent to the $x$-axis at $(x,0)$ and
adds the point of tangency. --- Show that the $x$-axis is a closed
subset and has the discrete topology.

(b) Explain why the space in question is Hausdorff.  [{\it Hint: }
The topology contains the metric topology.  If a topological space
is Hausdorff and we take a larger topology, why is the new
topology Hausdorff?]

(c) Show that the set of points $(u,v)$  in $X$ with $v>0$ and 
$u,v\in {\bf Q}$ is dense.  [{\it Hint: } Reduce this to showing
that one can find such a point in every set of the form 
$T_\varepsilon(x)$.]
\medskip
{\bf 3. } \quad 
Let $\{ A_\alpha~|~\alpha \in 
{\cal A} \}$ be a family of topological spaces, and let
$X = \coprod_\alpha A_\alpha$.  Formulate and prove necessary and
sufficient conditions on $\cal A$ and the sets $A_\alpha$ for
the space $X$ to be second countable, separable or Lindel\"of.
\vfil\eject
\bigskip\bigskip
\centerline{\bf VI.2$\;$:\quad Compact spaces -- II}
\bigskip
\centerline{(Munkres, \S\S~26, 27,28)}
\medskip
Munkres, \S~28, pp. 181--182:~6
%\vfil\eject
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1.$\ist$ } \quad
Let $X$ be a compact Hausdorff space, let $Y$ be a Hausdorff space, 
and let $f: X \rightarrow Y$ be a
continuous map such that $f$ is locally 1--1 (each point $x$ has a
neighborhood $U_x$ such that $f|U_x$ is 1--1) and there is a closed
subset $A \subset X$ such that $f|A$ is 1--1.  Prove that there is an
open neighborhood $V$ of $A$ such that $f|V$ is 1--1.  [{\it Hint:} A
map $g$ is 1--1 on a subset $B$ is and only if 
$$B \times B \cap (g \times g)^{-1} (\Delta_Y) = \Delta_B,$$
where $\Delta_S$ denotes the diagonal in $S \times S$.  In the setting
of the exercise show that 
$$(f \times f)^{-1} ( \Delta_Y ) = \Delta_X \cup D' ,$$
where $D'$ is closed and disjoint from the diagonal.  Also show that the
subsets $D'$ and $A \times A$ are disjoint, and find a square
neighborhood of $A \times A $ disjoint from $D'$.]
\medskip
{\bf 2. } \quad 
Let $U$ be open in ${\bf R}^n$, and let $f:U\to {\bf R}^n$ be
a ${\bf C}^1$ map such that $Df(x)$ is invertible for all
$x\in U$ and there is a compact subset $A\subset U$ such that $f|A$ is
1--1.  Prove that there is an open neighborhood  $V$ of $A$ 
such that $f|V$ is a homeomorphism onto its image.
\medskip
{\bf 3.$\iist$ } \quad 
Let ${\bf d}_p$ be the metric on the integers constructed in
Exercise I.1.1, and let ${\widehat {{\bf Z}_p}}$ be the completion
of this metric space.  Prove that ${\widehat {{\bf Z}_p}}$ is
(sequentially) compact.  [{\it Hint: }  For each integer $r>0$ show
that every integer is within $p^{-r}$ of one of the first
$p^{r+1}$ nonnegative integers.  Furthermore, each open
neighborhood of radius $p^{-r}$ centered at one of these
integers $a$ is a union of $p$ neighborhoods of radius
$p^{-(r+1)}$ over all of the first $p^{r+2}$ integers $b$
such that $b \equiv a$ mod $p^{r+1}$.  Now let $\{\,a_n\,\}$ be an 
infinite sequence of integers, and assume the sequence takes
infinitely many distinct values (otherwise the sequence obviously
has a convergent subsequence).  Find a sequence of positive integers 
$\{\,b_r\,\}$ such that the open neighborhood of radius $p^{-r}$ 
centered at $b_r$ contains infinitely many points in the
sequence and $b_{r+1}\equiv b_r$ mod $p^{r+1}$.  Prove that this
yields a Cauchy sequence in {\bf Z} (with respect to the 
${\bf d}_p$ metric).  Form a subsequence of $\{\,a_n\,\}$ by
choosing distinct points $a_{n(k)}$ recursively such that 
$n(k)>n(k-1)$ and $a_{n(k)}\in N_{p^{-k}}(b_k)$.
Prove that this
subsequence is a Cauchy sequence and hence converges.]
\bigskip\bigskip
\centerline{\bf VI.3$\;$:\quad Separation axioms}
\bigskip
\centerline{(Munkres, \S\S~31, 32, 33, 35)}
\medskip
Munkres, \S~26, pp. 170--172:~11$\ist$

Munkres, \S~33, pp. 212--214:~2(a) (for metric spaces), 6(a), 8
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad
If $(X,{\bf T})$ is compact Hausdorff and ${\bf T}^*$ is strictly
contained in {\bf T}, prove that $(X,{\bf T}^*)$ is compact but
not Hausdorff.
\medskip
{\bf 2. } \quad
(a)  Prove that a topological space is ${\bf T}_{\bf 3}$ if and only
if it is ${\bf T}_{\bf 1}$ and there is a basis $\cal B$ such that 
for every $x\in X$ and every open set $V\in {\cal B}$ containing
$x$, there is an open subset $W\in {\cal B}$ such that
$x\in W\subset {\overline W} \subset V$.

(b)$\iist$ Prove that the space constructed in Exercise VI.1.2 is 
${\bf T}_{\bf 3}$.  [{\it Hint: } Remember that the ``new'' topology
contains the usual metric topology.]
\medskip
{\bf 3. } \quad
If $X$ is a topological space and $A\subset X$ is nonempty then
$X/A$ (in words, ``$X$ mod $A$'' or ``$X$ modulo $A$ collapsed to
a point'') is the quotient space whose
equivalence classes are $A$ and all one point subsets $\{x\}$ such
that $x\not\in A$.  Geometrically, one is collapsing $A$ to a 
single point.  

(a) Suppose that $A$ is closed in $X$.  Prove that 
$X/A$ is Hausdorff if either $X$ is compact Hausdorff or $X$ is 
metric (in fact, if $X$ is 
${\bf T}_{\bf 4}$).

(b) Still assuming $A$ is closed but not making any assumptions
on $X$ (except that it be nonempty), show that the quotient map
$X\to X/A$ is always closed but not necessarily open.
[{\it Note: } For reasons that we shall not
discuss, it is appropriate to define $X/\emptyset$ to be the
disjoint union $X\sqcup \{\emptyset\}$.]
%\vfil\eject
\bigskip\bigskip
\centerline{\bf VI.4$\;$:\quad Local compactness and compactifications}
\bigskip
\centerline{(Munkres, \S\S~29, 37, 38)}
\medskip
Munkres, \S~38, pp. 241--242:~2$\ist$, 3 (just give a necessary condition 
on the topology of the space)
\medskip
\centerline{\it Additional exercises}
\bigskip
\noindent{\bf Definition. } If $f:X\to Y$ is continuous, then $f$
is {\it proper} (or {\it perfect}) if for each compact subset
$K\subset Y$ the inverse image $f^{-1}(K)$ is a compact
subset of $X$.
\medskip
{\bf 1. } \quad
Suppose that $f: X \rightarrow Y$ is a continuous map of
noncompact locally compact $T_2$ spaces.  Let $f^\bullet : X^\bullet
\rightarrow Y^\bullet$ be the map of one point comapactifications
defined by $f^\bullet | X = f$ and $f^\bullet ( \infty_X ) = 
(\infty_Y )$.  Prove that $f$ is proper if and only if $f^\bullet$ is
continuous.
\medskip
{\bf 2. } \quad
Prove that a proper map of noncompact locally compact Hausdorff
spaces is closed.
\medskip
{\bf 3. } \quad
If ${\bf F}$ is the reals or complex numbers, prove that every
polynomial map 
$p:{\bf F}\to{\bf F}$ is proper.  [{\it Hint: }  Show that
$$\lim_{|x|\to\infty}\; \bigl|p(z)\bigr|~~ =~~ \infty$$
and use the characterization of compact subsets as closed and
bounded subsets of {\bf F}.]
\medskip
{\bf 4.$\ist$ } \quad
Let ${\ell}^2$ be the complete metric space described above, 
and view ${\bf R}^n$ as the subspace of all 
sequences with $x_k = 0$ for $k > n$.  Let $A_n \subset {\ell}^2 \times 
{\bf R}$ be the set of all ordered 
pairs $(x,t)$ with $x \in {\bf R}^n$ and $0 < t \leq 2^{-n}$.  Show that 
$A = \bigcup_n A_n$ is locally compact but its closure is not.  Explain why 
this shows that the completion of a locally compact metric space is not
neccesarily locally compact.  [{\it Hint:} The family $\{\,A_n\,\}$ is 
a locally finite family of closed locally compact subspaces in $A$.  Use 
this to show that the union is locally compact, and show that the closure 
of $A$ contains all of ${\ell}^2 \times \{ 0 \}$.  Explain why 
${\ell}^2$ is not locally compact.]
\medskip
{\bf 5. } \quad
Let $X$ be a compact Hausdorff space, and let $U\subset X$ be open
and noncompact.  Prove that the collapsing map $c:X\to U^\bullet$
such that $c|U = {\rm id}_U$ and $c = \infty_U$ on $X-U$ is 
continuous.  Show also that $c$ is not necessarily open.
\medskip
{\bf 6. } \quad
(a) Explain why a compact Hausdorff space has no nontrivial
Hausdorff abstract closures.

(b) Prove that a Hausdorff space $X$ has a maximal abstract 
Hausdorff closure that is unique up to equivalence.  [{\it Hint: }
Consider the identity map.]
\medskip
{\bf 7. } \quad
Suppose that $X$ is compact Hausdorff and $A$ is a closed subset
of $X$.  Prove that $X/A$ is homeomorphic to the one point
compactification of $X-A$.
\bigskip\bigskip
\centerline{\bf VI.5$\;$:\quad Metrization theorems}
\bigskip
\centerline{(Munkres, \S\S~39, 40, 41, 42)}
\medskip
Munkres, \S~40, p. 252:~2, 3
\medskip
\centerline{\it Additional exercises}
\medskip
{\bf 1. } \quad 
A {\it pseudometric space} is a pair $(X,{\bf d})$ consisting of
a nonempty set $X$ and a function ${\bf d}X\times X\to {\bf R}$ 
that has all the properties of a metric except possibly the 
property that ${\bf d}(u,v) = 0$.  

(a)  If $\varepsilon$-neighborhoods and open sets are defined 
as for metric spaces, explain why one still obtains a topology
for pseudometric spaces.  

(b) Given a pseudometric space, define a binary relation 
$x\sim y$ if and only if ${\bf d}(x,y) = 0$.  Show that this
defines an equivalence relation and that ${\bf d}(x,y)$
only depends upon  the equivalence classes of $x$ and $y$.

(c) Given a sequence of pseudometrics ${\bf d}_n$ on a set $X$,
let ${\bf T}_\infty$ be the topology generated by the union of 
the sequence of topologies associated to these pseudometrics, 
and suppose that for each pair of distinct points $u\,v\in X$ 
there is some $n$ such that ${\bf d}_n(u,v)>0$.  Prove that
$(X,{\bf T}_\infty)$ is metrizable and that
$${\bf d}_\infty ~~=~~
\sum_{n=1}^\infty {{{\bf d}_n}\over{2^n(1+{\bf d}_n)}}$$
defines a metric whose underlying topology is ${\bf T}_\infty$.

(d) Let $X$ be the set of all continuous real valued functions on
the real line {\bf R}.  Prove that $X$ is metrizable such
that the restriction maps from $X$ to
${\bf BC}(\,[-n,n]\,)$ are uniformly continuous for all $n$.  
[{\it Hint: } Let ${\bf d}_n (f,g)$ be the maximum value of 
$|f(x) - g(x)|$ for $|x|\leq n$.]

(e) Given $X$ and the metric constructed in the previous
part of the problem, prove that a sequence of functions
$\{\,f_n\,\}$ converges to $f$ if and only if for each
compact subset $K\subset {\bf R}$ the sequence of 
restricted functions $\{\,f_n|K\,\}$ converges to 
$f|K$.

(f) Is $X$ complete with respect to the metric described above?  
Prove this or give a counterexample.

(g) Explain how the preceding can be generalized from
continuous functions on {\bf R} to continuous functions on 
an arbitrary open subset $U\subset {\bf R}^n$.
\medskip
{\bf 2. } \quad 
(a) Let $U$ be an open subset of ${\bf R}^n$, and let
$f:U\to {\bf R}^n$ be a continuous function such that
$f^{-1}(\{0\})$ is contained in an open subset $V$
such that $V\subset {\overline V} \subset U$.  Prove
that there is a continuous function $g$ from $S^n \cong 
({\bf R}^n)^\bullet$ to itself such that $g|V = 
f|V$ and $f^{-1}(\{0\}) = g^{-1}(\{0\})$. [{\it Hint: }
Note that 
$$({\bf R}^n)^\bullet~-~\{0\}~~\cong~~{\bf R}^n$$
and consider the continuous function on 
$$\left({\overline V}-V\right)~\sqcup \{\infty\}~~
\subset~~({\bf R}^n)^\bullet~-~\{0\}$$
defined on the respective pieces by the restriction of $f$
and $\infty$.  Why can this be extended to a continuous 
function on 
$({\bf R}^n)^\bullet~-~V$ with the same codomain?  What
happens if we try to piece this together with the original
function $f$ defined on $U$?]

(b) Suppose that we are given two continuous functions $g$ and
$g'$ satisfying the conditions of the first part of this
exercise.  Prove that there is a continuous funtion
$$G:S^n\times[0,1]\longrightarrow S^n$$
such that $G(x,0)$ = $g(x)$ for all $x\in S^n$ and
$G(x,1)$ = $g'(x)$ for all $x\in S^n$ ({\it i.e.}, the
mappings $g$ and $g'$ are {\bf homotopic}).

\medskip
{\bf 3. } \quad 
Let $X$ be compact, and let ${\cal F}$ be a family of continuous real 
valued functions on $X$ that is closed under multiplication and 
such that for each $x \in X$ there is a neighborhood $U$ of $X$ and a 
function $f \in {\cal F}$ that vanishes identically on $U$.  Prove 
that ${\cal F}$ contains the zero function.
\medskip
{\bf 4.$\iist$ } \quad 
Let $X$ be a compact metric space, and let $J$ be a nonempty subset
of the ring ${\bf BC}(X)$ of (bounded) continuous functions on $X$
such that $J$ is closed under addition and subtraction, it is an
{\it ideal} in the sense that $f\in J$ and $g\in {\bf BC}(X)
\Longrightarrow f\cdot g\in J$, and for each $x\in X$ there is
a function $f\in J$ such that $f(x)\neq 0$.  Prove that 
$J = {\bf BC}(X)$.  [{\it Hints: }  This requires the 
existence of partitions of unity  as established in Theorem 36.1
on pages 225--226 of Munkres; as noted there, the result works for 
arbitrary compact Hausdorff spaces, but we restrict to metric
spaces because the course does not cover Urysohn's Lemma in
that generality.  Construct a finite open covering of $X$, say 
$\cal U$, such that for each $U_i\in {\cal U}$ there is a 
function $f_i\in J$ such that $f_i>0$ on $U_i$.  Let $\{\,\varphi_i
\,\}$ be a partition of unity dominated by $\cal U$, and
form $h = \sum_i \varphi_i\cdot f_i$.  Note that $h\in J$ and $h>0$
everywhere so that $h$ has a reciprocal $k = 1/h$ in ${\bf BC}(X)$.
Why does this imply that the constant function 1 lies in $J$, and
why does the latter imply that everything lies in $J$?]
\medskip
{\bf 5.$\iist$ } \quad 
In the notation of the preceding exercise, an ideal ${\bf M}$
in ${\bf BC}(X)$ is said to be a {\it maximal ideal} if it is
a proper ideal and there are no ideals ${\bf A}$ such that
${\bf M}$ is properly contained in ${\bf A}$ and ${\bf A}$ is
properly contained in in ${\bf BC}(X)$.  Prove that there is a 1--1
correspondence between the maximal ideals of ${\bf BC}(X)$ 
and the points of $X$ such that the ideal ${\bf M}_x$
corresponding to $X$ is the set of all continuous functions 
$g:X\to {\bf R}$ such that $g(x) = 0$.  [{\it Hint: }  Use
the preceding exercise.]
\bigskip\bigskip
\vfil\eject
%%%%%%%%%%%%%%%%%%%%
\centerline{\Head APPENDICES}
\bigskip\bigskip
\centerline{\bf Appendix A$\;$: Topological groups}
\bigskip
\bigskip
\centerline{(Munkres, Supplementary exercises following \$~22; {\it see
also} course notes, Appendix D)}
\bigskip
\bigskip
Munkres, \S~26, pp. 170--172:~12, 13
\medskip
Munkres, \S~30, pp. 194--195:~18
\medskip
Munkres, \S~31, pp. 199--200:~8
\medskip
Munkres, \S~33, pp. 212--214:~10
\medskip
\centerline{\it Additional exercises}
\medskip
Let {\bf F} be the real or complex numbers.  Within the matrix group
${\bf GL}(n,{\bf F})$ there are certain subgroups of particular importance. 
One such subgroup is the {\it special linear group} ${\bf SL}(n,{\bf F})$ 
of all matrices of determinant 1.  
\medskip
{\bf 0. } \qquad  
Prove that the group ${\bf SL}(2,{\bf C})$ has no nontrivial proper
normal subgroups except for the subgroup
$\{\pm\,I\}$.  [{\it Hint: }  If $N$ is a normal subgroup, 
show first thatif $A\in N$ then $N$ contains all matrices that
are similar to $A$.  Therefore the proof reduces to considering
normal subgroups containing a Jordan form matrix of one of
the following two types:
$$
\left(
\matrix{
\alpha&0\cr
0&\alpha^{-1}
}\right)\quad,\qquad
\left(
\matrix{
\varepsilon&1\cr
0&\varepsilon
}\right)$$
Here $\alpha$ is a complex number not equal to 0 or $\pm\,1$ and
$\varepsilon = \pm\,1$.  The idea is to show that if $N$
contains one of these Jordan forms then it contains all such
forms, and this is done by computing sufficiently many matrix 
products.  Trial and error is a good way to approach this
aspect of the problem.]
\medskip
\noindent{\bf Definition. }
The {\it orthogonal group} ${\bf O}(n)$ consists of all 
transformations in ${\bf GL}(n,{\bf R})$ that take each orthonormal 
basis for ${\bf R}^n$ to another orthonormal basis, or equivalently,
the subgroup of all matrices whose columns form an orthonormal basis.
It is an easy exercise in linear algebra to show that the determinant of all 
matrices in ${\bf O}(n)$ is $\pm\,1$. The {\it special orthogonal group} 
${\bf SO}(n)$ is the subgroup of ${\bf O}(n)$ consisting
of all matrices whose determinants are equal to  $+1$.
Replacing 
If we replace the real numbers {\bf R} by the complex numbers {\bf C} 
we get the unitary groups ${\bf U}(n)$ and the {\it special
unitary groups} ${\bf SU}(n)$, which are the subgroups of ${\bf U}(n)$ 
given by matrices with determinant 1.
The determinant of every matrix in ${\bf U}(n)$ is of absolute value 1 
just as before, but in the complex case this means that the 
determinant is a complex number on the unit circle.
In Appendix A the orthogonal and unitary groups were shown to be
compact.
\medskip
{\bf 1. } \qquad  
Show that ${\bf O}(1)$ is isomorphic to the cyclic group of order 2 
and that ${\bf SO)}(2)$ is isomorphic as a topological group to the 
circle group $S^1$. Conclude that ${\bf O}(2)$ is homeomorphic as 
a space to ${\bf SO}(2) \times {\bf O}(1)$, but that as a group these
objects are not isomorphic to each other.  [{\it Hint: } In Appendix
D it is noted that every element of ${\bf O}(2)$ that is not in
${\bf SO}(2)$ has an orthonormal basis of eigenvectors
corresponding to the eigenvalues $\pm\,1$.  What does this say
about the orders of such group elements?]
\medskip
{\bf 2. } \quad
Show that ${\bf U}(1)$ is isomorphic to the circle group $S^1$ as a 
topological group.
\medskip
{\bf 3. } \quad
For each positive integer $n$, show that ${\bf SO}(n)$,
${\bf U}(n)$ and ${\bf SU}(n)$ are connected spaces and that 
${\bf U}(n)$ is homeomorphic as a space (but not necessarily
isomorphic as a topological group) to ${\bf SU}(n) \times S^1$.
[{\it Hints: } In the complex case use the Spectral Theorem for
normal matrices to show that every unitary matrix lies in the
path component of the identity, which is a normal subgroup.
In the real case use the results on normal form in Appendix D.]
\medskip
{\bf 4. } \quad
Show that ${\bf O}(n)$ has two connected components both homeomorphic 
to ${\bf SO}(n)$ for every $n$.
\medskip
{\bf 5. } \quad
Show that the inclusions of ${\bf O}(n)$ in ${\bf GL}(n,{\bf R})$ 
and ${\bf U}(n)$ in ${\bf GL}(n,{\bf C})$ 
determine 1--1 correspondences of path components.
[{\it Hint: } Show that the Gram-Schmidt orthonormalization
process expresses an arbitrary invertible matrix over the
real numbers as a product $PQ$ where $P$ is upper
triangular and $Q$ is orthogonal (real case) or unitary
(complex case), and use this to show that every invertible
matrix can be connected to an orthogonal or unitary matrix
by a continuous curve.]
\bye

