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.