Equivalent conditions to a map being a quotient map

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:

Review and demote

Note: this page will have to be re-written if other conditions are to be added.

Statement

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $f:X\rightarrow Y$ be a map. Then:

• $f$ is a quotient map

if and only if

• It does either one, or both, of the following:
  1. $f$ takes saturated open sets to open sets
  2. $f$ takes saturated closed sets to closed sets

Proof

Grade: A

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:

Unfortunately left as an exercise