Completely regular space

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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 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 mapf: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
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. MunkresMore 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. ThorpeMore info, Page 37 (formal definition)