Glossary of Topology - T

A space is T0 (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.
A space is T1 (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 T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. Every T1 space is T0.
See Hausdorff space.
See Regular Hausdorff.
See Tychonoff space.
See Normal Hausdorff.
See Completely normal Hausdorff.
See Category of topological spaces.
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 axioms:
  1. The empty set and X are in T.
  2. The union of any collection of sets in T is also in T.
  3. 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
A space is topologically complete if it is homeomorphic to a complete metric space.
See Topological space.
Totally bounded
A metric space M is totally bounded if, for every r > 0, there exist a 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.
A Tychonoff space (or completely regular Hausdorff space, completely T3 space, T3.5 space) is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

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