Editing Glossary of general topology (section)

== F ==

;[[F-sigma set|''F''<sub>σ</sub> set]]: An [[F-sigma set|''F''<sub>σ</sub> set]] is a [[countable]] union of closed sets.<ref name=ss162/>

;[[Filter (set theory)|Filter]]: See also: [[Filters in topology]]. A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold:
:# The [[empty set]] is not in ''F''.
:# The intersection of any [[finite set|finite]] number of elements of ''F'' is again in ''F''.
:# If ''A'' is in ''F'' and if ''B'' contains ''A'', then ''B'' is in ''F''.

;[[Final topology]]: On a set ''X'' with respect to a family of functions into <math>X</math>, is the [[finest topology]] on ''X'' which makes those functions [[continuous function (topology)|continuous]].<ref>{{cite book | first=Stephen | last=Willard | title=General Topology | url=https://archive.org/details/generaltopology00will_0 | url-access=registration | year=1970 | publisher=Addison-Wesley | location=Reading, MA | zbl=0205.26601 | series=Addison-Wesley Series in Mathematics| isbn=9780201087079 }}</ref>

;[[Fine topology (potential theory)]]: On [[Euclidean space]] <math>\R^n</math>, the coarsest topology making all [[subharmonic function]]s (equivalently all superharmonic functions) continuous.<ref>{{cite book | first=John B. | last=Conway | author-link=John B. Conway | isbn=0-387-94460-5 | series=[[Graduate Texts in Mathematics]] | volume=159 | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | year=1995 | zbl=0887.30003 | pages=367–376 }}</ref>

;[[Finer topology]]: If ''X'' is a set, and if ''T''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>2</sub> is [[finer topology|finer]] (or '''larger''', '''stronger''') than ''T''<sub>1</sub> if ''T''<sub>2</sub> contains ''T''<sub>1</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''weaker'''.

;Finitely generated: See '''[[Alexandrov topology]]'''.

;[[First category]]: See '''[[Meagre set|Meagre]]'''.

;[[First-countable space|First-countable]]: A space is [[First-countable space|first-countable]] if every point has a [[countable]] local base.

;Fréchet: See '''[[T1 space|T<sub>1</sub>]]'''.

;Frontier: See '''[[Boundary (topology)|Boundary]]'''.

;Full set: A [[compact space|compact]] subset ''K'' of the [[complex plane]] is called '''full''' if its [[complement (set theory)|complement]] is connected. For example, the [[closed unit disk]] is full, while the [[unit circle]] is not.

;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X'' &nbsp;→&nbsp; [0, 1] such that ''f''(''A'') = 0 and ''f''(''B'') = 1.