Editing Glossary of general topology (section)

== T ==

;[[T0 space|T<sub>0</sub>]]: A space is [[T0 space|T<sub>0</sub>]] (or '''Kolmogorov''') if for every pair of distinct points ''x'' and ''y'' in the space, either there is an open set containing ''x'' but not ''y'', or there is an open set containing ''y'' but not ''x''.

;[[T1 space|T<sub>1</sub>]]: A space is [[T1 space|T<sub>1</sub>]] (or '''Fréchet''' or '''accessible''') if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T<sub>0</sub>; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T<sub>1</sub> if all its [[singleton (mathematics)|singleton]]s are closed. Every T<sub>1</sub> space is T<sub>0</sub>.

;[[T2 space|T<sub>2</sub>]]: See '''[[Hausdorff space]]'''.

;[[T3 space|T<sub>3</sub>]]: See '''[[T3 space|Regular Hausdorff]]'''.

;[[Tychonoff space|T<sub>3½</sub>]]: See '''[[Tychonoff space]]'''.

;[[T4 space|T<sub>4</sub>]]: See '''[[T4 space|Normal Hausdorff]]'''.

;[[T5 space|T<sub>5</sub>]]: See '''[[T5 space|Completely normal Hausdorff]]'''.

;[[Category of topological spaces|Top]]: See '''[[Category of topological spaces]]'''.

;θ-cluster point, θ-closed, θ-open: A point ''x'' of a topological space ''X'' is a θ-cluster point of a subset ''A'' if <math>A \cap \operatorname{Cl}_X(U) \neq \emptyset</math> for every open neighborhood ''U'' of ''x'' in ''X''.  The subset ''A'' is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed.{{sfn|Hart|Nagata|Vaughan|2004|p=8}}

;[[Topological invariant]]: A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. [[Algebraic topology]] is the study of topologically invariant [[abstract algebra]] constructions on topological spaces.

;[[Topological space]]: A [[topological space]] (''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following [[axiom]]s:

:# The empty set and ''X'' are in ''T''.
:# The union of any collection of sets in ''T'' is also in ''T''.
:# The intersection of any pair of sets in ''T'' is also in ''T''.

:The collection ''T'' is a '''topology''' on ''X''.

;Topological sum: See '''Coproduct topology'''.

;Topologically complete: [[Completely metrizable space]]s (i. e. topological spaces homeomorphic to complete metric spaces) are often called ''topologically complete''; sometimes the term is also used for [[Čech-complete space]]s or [[completely uniformizable space]]s.

;Topology: See '''[[Topological space]]'''.

;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a [[finite set|finite]] cover of ''M'' by open balls of radius ''r''. A metric space is compact if and only if it is complete and totally bounded.

;Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point.

;[[Trivial topology]]: The [[trivial topology]] (or '''indiscrete topology''') on a set ''X'' consists of precisely the empty set and the entire space ''X''.

;[[Tychonoff space|Tychonoff]]: A [[Tychonoff space]] (or '''completely regular Hausdorff''' space, '''completely T<sub>3</sub>''' space, '''T<sub>3.5</sub>''' space) is a completely regular T<sub>0</sub> space.  (A completely regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.