0708-1300/Class notes for Thursday, November 1

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Announcements go here

Today's Agenda

• HW4 and TE1.
• Continue with Tuesday's agenda:
  • Debt on proper functions.
  • Prove that "the sphere is not contractible".
  • Complete the proof of the "tubular neighborhood theorem".

Proper Implies Closed

Theorem. A proper function $f:X\to Y$ from a topological space $X$ to a locally compact (Hausdorff) topological space $Y$ is closed.

Proof. Let $B$ be closed in $X$, we need to show that $f(B)$ is closed in $Y$. Since closedness is a local property, it is enough to show that every point $y\in Y$ has a neighbourhood $U$ such that $f(B)\cap U$ is closed in $U$. Fix $y\in Y$, and by local compactness, choose a neighbourhood $U$ of $y$ whose close ${\bar {U}}$ is compact. Then

$f(B)\cap U=f(B\cap f^{-1}(U))\cap U\subset f(B\cap f^{-1}({\bar {U}}))\cap U\subset f(B)\cap U$,

so that $f(B)\cap U=f(B\cap f^{-1}({\bar {U}}))\cap U$. But ${\bar {U}}$ is compact by choice, so $f^{-1}({\bar {U}})$ is compact as $f$ is proper, so $B\cap f^{-1}({\bar {U}})$ is compact as $B$ is closed, so $f(B\cap f^{-1}({\bar {U}}))$ is compact (and hence closed) as a continuous image of a compact set, so $f(B)\cap U$ is the intersection $f(B\cap f^{-1}({\bar {U}}))\cap U$ of a closed set with $U$, hence it is closed in $U$.

Note

The example of a non-contractible "comb" seen today is, in fact, "Cantor's comb". See, for example, page 25 of www.karlin.mff.cuni.cz/~pyrih/e/e2000v0/c/ect.ps