Final exam - Math 671 - Topology - Fall 2000 - Rob Kusner

Work alone.
Answer all questions.
Justify your answers.

Return completed exam papers to Rob (or his mailbox) before 23 December 2000.



1.  Consider a few of the ``other'' topologies on $\R$: 

	(i) trivial, 

	(ii) Zariski   ( = finite complement),

	(iii) discrete. 

For each of these topologies check whether the interval $[a,b]$ is

	(a) closed,    (b) connected,   (c)  compact?


2.  Show that every map from the circle $S^1 \ra\ \R^n$ extends to a
map from the disk $D^2 \ra\ \R^n$.


3.  Consider the ``infinite tee'' in the plane, i.e., the subspace
$T\subset\R^2$ defined by the union of the $x$-axis and the negative
$y$-axis.  Show that $T$ has a one-point compactification $T'$.  To
which letter of the Greek alphabet is $T'$ homeomorphic?


4.  Decide whether the subspaces $A$, $B$ and $C$ of $\R^2$ suggested
by the corresponding letters of the Roman alphabet, are homeomorphic
to each other or not.  If a pair of spaces is not homeomorphic, can
one be embedded in the other?


5.  Let $X$ and $Y$ be connected and let $A\subset X$ and $B\subset Y$
be proper subsets. Show that $(X\tim Y) - (A\tim B)$ is connected.


6. Let $X$ be any space, and let $Y$ be a compactly generated,
Hausdorff space.  Show that a proper bijective map $$f: X \ra Y$$ is a
homeomorphism.