Tychonoff space

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Definition

A topological space is termed a Tychonoff space if it satisfies the following equivalent conditions:

No. Shorthand A topological space is termed Tychonoff if ... A topological space X is termed Tychonoff if ...
1 T1 and completely regular it is both a T1 space and a completely regular space points are closed in X, and given any point x \in X and closed subset A \subseteq X such that x \notin A, there exists a continuous mapf:X \to [0,1] such that f(x) = 0 and f(a) = 1 for all a \in A.
2 Hausdorff and completely regular it is both a Hausdorff space and a completely regular space it is Hausdorff, and given any point x \in X and closed subset A \subseteq X such that x \notin A, there exists a continuous mapf:X \to [0,1] such that f(x) = 0 and f(a) = 1 for all a \in A.
3 has a compactification there is a compact Hausdorff space having a dense subspace (with the subspace topology) homeomorphic to it. (note: T1 assumption redundant in this case) there is a compact Hausdorff space C and a dense subspace U of X such that X is homeomorphic to U.
4 contained in compact Hausdorff it is homeomorphic to a subspace (not necessarily dense) of a compact Hausdorff space (note: T1 assumption redundant in this case). there is a compact Hausdorff space C and a subspace U of X such that X is homeomoephic to U.
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

In the T family (properties of topological spaces related to separation axioms), this is called: T3.5

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal Hausdorff space Hausdorff and a normal space: disjoint closed subsets can be separated via disjoint open subsets |FULL LIST, MORE INFO
compact Hausdorff space Hausdorff and a compact space: every open cover has a finite subcover via compact Hausdorff implies normal Paracompact Hausdorff space|FULL LIST, MORE INFO
underlying space of T0 topological group |FULL LIST, MORE INFO
metrizable space underlying topological space of a metric space Paracompact Hausdorff space|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
completely regular space |FULL LIST, MORE INFO
regular Hausdorff space |FULL LIST, MORE INFO
regular space Regular Hausdorff space|FULL LIST, MORE INFO
Hausdorff space Functionally Hausdorff space, Regular Hausdorff space, Urysohn space|FULL LIST, MORE INFO