The image of a compact set is compact

Stub grade: A

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Flesh out with references, proof is small but easy and can wait

Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath] and let [ilmath]f:X\rightarrow Y[/ilmath] be a continuous map. Then:

• if [ilmath]A[/ilmath] is compact then [ilmath]f(A)[/ilmath] is compact.

Grade: A

This page requires references, it is on a to-do list for being expanded with them.

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I want Mendelson. Also should I make a note that considered as a subspace or compactness-as-a-subset are the same?

Proof

• Let [ilmath]\mathcal{U} [/ilmath] be an open cover of [ilmath]f(A)[/ilmath] and then showing it has a finite subcover
  • The proof depends on [ilmath]\{f^{-1}(U)\ \vert\ U\in\mathcal{U} \} [/ilmath] being an open cover of [ilmath]A[/ilmath] by sets open in [ilmath](X,\mathcal{ J })[/ilmath]
    1. The open part comes from the [ilmath]U[/ilmath] be open sets, then by the definition of continuity [ilmath]f^{-1}(U)[/ilmath] is open in [ilmath]X[/ilmath] for each [ilmath]U\in\mathcal{U} [/ilmath]
    2. We need to show [math]A\subseteq\bigcup_{U\in\mathcal{U} }f^{-1}(U)[/math], by the implies-subset relation and then the definition of union we see:
       • [math]\left(A\subseteq\bigcup_{U\in\mathcal{U} }f^{-1}(U)\right)\iff\forall x\in A\left[x\in\bigcup_{U\in\mathcal{U} }f^{-1}(U)\right][/math] [math]\iff\forall x\in A\exists U\in\mathcal{U}[x\in f^{-1}(U)][/math]
       • In this case [ilmath]\{f^{-1}(U)\ \vert\ U\in\mathcal{U} \} [/ilmath] is an open cover of [ilmath]A[/ilmath]
  • Then we see, by compactness of [ilmath]A[/ilmath]:
    • there is some [ilmath]n\in\mathbb{N} [/ilmath] such that [ilmath]\{f^{-1}(U_1),\ldots,f^{-1}(U_n)\} [/ilmath]

Grade: C

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

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The message provided is:

Proof isn't that important as it is easy and routine.

• Done a chunk, but not finished. Alec (talk) 17:20, 18 December 2016 (UTC)