# 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

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## Definition

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

No. Shorthand A topological space is termed completely regular if ... A 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 $X$.

### Convention issues

Note that in some conventions, the $T_1$ assumption is made along with completely regular. We use the term Tychonoff space here for a completely regular space that is also $T_1$.

## 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
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.