Tychonoff space

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 map$f:X\to [0,1]$ 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 map$f:X\to [0,1]$ 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