The image of a compact set is compact

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

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

Statement

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

• if $A$ is compact then $f(A)$ is compact.

Grade: A

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

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:

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

Proof

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

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.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
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)