Editing F-space (section)

== Related properties ==

The [[Open mapping theorem (functional analysis)|open mapping theorem]] implies that if <math>\tau \text{ and } \tau_2</math> are topologies on <math>X</math> that make both <math>(X, \tau)</math> and <math>\left(X, \tau_2\right)</math> into [[Complete topological vector space|complete]] [[metrizable topological vector space]]s (for example, Banach or [[Fréchet space]]s) and if one topology is [[Comparison of topologies|finer or coarser]] than the other then they must be equal (that is, if <math>\tau \subseteq \tau_2 \text{ or } \tau_2 \subseteq \tau \text{ then } \tau = \tau_2</math>).{{sfn|Trèves|2006|pp=166–173}}

* A linear [[almost continuous]] map into an F-space whose graph is closed is continuous.{{sfn|Husain|Khaleelulla|1978|p=14}}
* A linear [[Almost open map|almost open]] map into an F-space whose graph is closed is necessarily an [[open map]].{{sfn|Husain|Khaleelulla|1978|p=14}}
* A linear continuous [[Almost open map|almost open]] map from an F-space is necessarily an [[open map]].{{sfn|Husain|Khaleelulla|1978|p=15}}
* A linear continuous almost open map from an F-space whose image is of the [[second category]] in the codomain is necessarily a [[Surjection|surjective]] [[open map]].{{sfn|Husain|Khaleelulla|1978|p=14}}