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 from a topological space to a locally compact (Hausdorff) topological space is closed.
Proof. Let be closed in , we need to show that is closed in . Since closedness is a local property, it is enough to show that every point has a neighbourhood such that is closed in . Fix , and by local compactness, choose a neighbourhood of whose close is compact. Then
so that . But is compact by choice, so is compact as is proper, so is compact as is closed, so is compact (and hence closed) as a continuous image of a compact set, so is the intersection of a closed set with , hence it is closed in .
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