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injective proper maps to locally compact spaces are equivalently the closed embeddings

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Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

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Statement

Proposition

(injective proper maps to locally compact spaces are equivalently the closed embeddings)

Let

  1. XX be a topological space

  2. YY a locally compact topological space

  3. f:XYf \colon X \to Y be a continuous function.

Then the following are equivalent

  1. ff is an injective proper map,

  2. ff is a closed embedding of topological spaces.

Proof

In one direction, if ff is an injective proper map, then since proper maps to locally compact spaces are closed, it follows that ff is also closed map. The claim then follows since closed injections are embeddings, and since the image of a closed map is closed.

Conversely, if ff is a closed embedding, we only need to show that the embedding map is proper. So for CYC \subset Y a compact subspace, we need to show that the pre-image f 1(C)Xf^{-1}(C) \subset X is also compact. But since ff is an injection (being an embedding), that pre-image is just the intersection f 1(C)Cf(X)f^{-1}(C) \simeq C \cap f(X). By the nature of the subspace topology, this is compact if CC is.

Created on May 16, 2017 13:53:49 by Urs Schreiber (131.220.184.222)