Editing Open and closed maps (section)

===Competing definitions===

There are two different competing, but closely related, definitions of "{{em|open map}}" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets." 
The following terminology is sometimes used to distinguish between the two definitions. 

A map <math>f : X \to Y</math> is called a
* "'''{{em|Strongly open map}}'''" if whenever <math>U</math> is an [[Open set|open subset]] of the domain <math>X</math> then <math>f(U)</math> is an open subset of <math>f</math>'s [[codomain]] <math>Y.</math> 
* "'''{{em|{{visible anchor|Relatively open map}}}}'''" if whenever <math>U</math> is an open subset of the domain <math>X</math> then <math>f(U)</math> is an open subset of <math>f</math>'s [[Image (mathematics)|image]] <math>\operatorname{Im} f := f(X),</math> where as usual, this set is endowed with the [[subspace topology]] induced on it by <math>f</math>'s codomain <math>Y.</math>{{sfn|Narici|Beckenstein|2011|pp=225-273}} 

Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.  

:'''Warning''': Many authors define "open map" to mean "{{em|relatively}} open map" (for example, The Encyclopedia of Mathematics) while others define "open map" to mean "{{em|strongly}} open map". In general, these definitions are {{em|not}} equivalent so it is thus advisable to always check what definition of "open map" an author is using. 

A [[Surjective function|surjective]] map is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent. 
More generally, a map <math>f : X \to Y</math> is relatively open if and only if the [[Surjective function|surjection]] <math>f : X \to f(X)</math> is a strongly open map. 

Because <math>X</math> is always an open subset of <math>X,</math> the image <math>f(X) = \operatorname{Im} f</math> of a strongly open map <math>f : X \to Y</math> must be an open subset of its codomain <math>Y.</math> In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain. 
In summary,

:A map is strongly open if and only if it is relatively open and its image is an open subset of its codomain.

By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition. 

The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".