0708-1300/Class notes for Thursday, November 1

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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