Completely regular space

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

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

This article is about a basic definition in topology.
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Definition

A topological space is termed completely regular if it is a T1 space and satisfies the following equivalent conditions:

No.	Shorthand	A T1 topological space is termed completely regular if ...	A T1 topological space $X$ is termed completely regular if ...
1	continuous function separating point and closed subset	given any point and any closed subset, there is a continuous map from the topological space to the closed unit interval $[0,1]$ that takes the value $0$ at the point and $1$ at the closed subset.	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	uniform structure	it occurs as the underlying topological space of a uniform space.	there is a uniform space structure ${\mathcal {U}}$ on $X$.
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 $U$.

Convention issues

Note that in some conventions, the $T_{1}$ assumption is not made. In this case, we call a space completely regular if, given any point and any closed set not containing it, there is a continuous function taking the value $0$ at the point and $1$ everywhere on the closed subset. This latter notion is a weaker notion of completely regular

Formalisms

In terms of the subspace operator

This property is obtained by applying the subspace operator to the property: compact Hausdorff space

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal space	disjoint closed subsets can be separated by a continuous function to $[0,1]$	normal implies completely regular	completely regular not implies normal	|FULL LIST, MORE INFO
locally compact Hausdorff space	Hausdorff and locally compact	locally compact Hausdorff implies completely regular	completely regular not implies locally compact Hausdorff	|FULL LIST, MORE INFO
underlying space of T0 topological group	Occurs as the underlying topological space of a T0 topological group			Tychonoff space|FULL LIST, MORE INFO
metrizable space	arises from a metric space via the metric-induced topology	(via normal)	(via normal)	Monotonically normal space, Normal Hausdorff space, Paracompact Hausdorff space, Tychonoff space|FULL LIST, MORE INFO
CW-space	arises from a CW-complex	(via normal)	(via normal)	Normal Hausdorff space, Paracompact Hausdorff space|FULL LIST, MORE INFO
polyhedron	geometric realization of a simplicial complex	(via normal)	(via normal)	Paracompact Hausdorff space|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular space (also called $T_{3}$)	$T_{1}$, and disjoint open subsets separating point and disjoint closed subset	completely regular implies regular	regular not implies completely regular	|FULL LIST, MORE INFO
Urysohn space	continuous function to $[0,1]$ separating any two distinct points	completely regular implies Urysohn	Urysohn not implies completely regular	|FULL LIST, MORE INFO
Hausdorff space (also called $T_{2}$)	distinct points can be separated by disjoint open subsets	(via regular)	(via regular)	|FULL LIST, MORE INFO
T1 space	points are closed	by definition		|FULL LIST, MORE INFO

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subspace-hereditary property of topological spaces	Yes	complete regularity is hereditary	If $X$ is a completely regular space and $A$ is a subset of $X$, then $A$ is completely regular with the subspace topology.
product-closed property of topological spaces	Yes	complete regularity is product-closed	If $X_{i},i\in I$, is a family of completely regular spaces, the product space $\prod _{i\in I}X_{i}$ is also a completely regular space with the product topology.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 211, Chapter 4, Section 33 (formal definition)
• Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 37 (formal definition)