embedding of topological spaces in nLab
Article Images
Context
Topology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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fiber space, space attachment
Extra stuff, structure, properties
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Kolmogorov space, Hausdorff space, regular space, normal space
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sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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open subspaces of compact Hausdorff spaces are locally compact
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compact spaces equivalently have converging subnet of every net
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continuous metric space valued function on compact metric space is uniformly continuous
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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injective proper maps to locally compact spaces are equivalently the closed embeddings
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locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Contents
Idea
An embedding of topological spaces is a continuous function which is a homeomorphism onto its image.
Definition
Definition
(embedding of topological spaces)
Let and be topological spaces. A continuous function is called an embedding if
in its image factorization
with the image equipped with the subspace topology, we have that is a homeomorphism.
Properties
For proof see at Top this proposition.
Lemma
In , the pushout of a (closed/open) embedding along any continuous map ,
is again a (closed/open) embedding.
For proof see at subspace topology here.
Proof
In one direction, if is an injective proper map, then since proper maps to locally compact spaces are closed, it follows that 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 is a closed embedding, we only need to show that the embedding map is proper. So for a compact subspace, we need to show that the pre-image is also compact. But since is an injection (being an embedding), that pre-image is just the intersection .
To see that this is compact, let be an open cover of the subspace , hence, by the nature of the subspace topology, let be a set of open subsets of , which cover and with the restriction of to . Now since is closed by assumption, it follows that the complement is open and hence that
is an open cover of . By compactness of this has a finite subcover. Since restricting that finite subcover back to makes the potential element disappear, this restriction is a finite subcover of . This shows that is compact.
Last revised on February 2, 2021 at 12:13:58. See the history of this page for a list of all contributions to it.