Functionally Hausdorff space

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Definition

A topological space is termed a completely Hausdorff space or functionally Hausdorff space if it satisfies the following equivalent conditions:

  1. For any two points in it, there is a continuous function from the whole space to [0,1] that takes the value 0 at one point and 1 at the other.
  2. For any two points in it, there is a continuous function from the whole space to the reals that takes the value 0 at one point and 1 at the other.
  3. For any two points in it, there is a continuous function from the whole space to the reals that takes distinct values at the two points.
  4. For any two points in it, and any two specified distinct real numbers, there is a continuous function from the whole space to the reals that takes the two specified values at the two points.
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
This is a variation of Hausdorffness. View other variations of Hausdorffness

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal Hausdorff space T_1 and any two disjoint closed subsets are separated by disjoint open subsets normal Hausdorff implies functionally Hausdorff functionally Hausdorff not implies normal |FULL LIST, MORE INFO
Tychonoff space T_1 and completely regular: continuous function to [0,1] separating any point and disjoint closed subset Tychonoff implies functionally Hausdorff functionally Hausdorff not implies completely regular |FULL LIST, MORE INFO
metrizable space |FULL LIST, MORE INFO
submetrizable space either metrizable or can become metrizable upon passing to a coarser topology |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Urysohn space any two distinct points can be separated by disjoint open subsets whose closures are also disjoint |FULL LIST, MORE INFO
Hausdorff space distinct points separated by disjoint open subsets Urysohn space|FULL LIST, MORE INFO
T1 space points are closed (via Hausdorff) (via Hausdorff) Hausdorff space|FULL LIST, MORE INFO
Kolmogorov space points are distinguishable (via Hausdorff) (via Hausdorff) Hausdorff space|FULL LIST, MORE INFO

Facts

Any connected Urysohn space with at least two points is uncountable (more precisely, its cardinality must be at least that of the continuum). This follows from the fact that its image under any continuous function must be connected, and hence the Urysohn function separating two points must be surjective to [0,1]. For full proof, refer: Connected Urysohn implies uncountable

Metaproperties

Metaproperty name Satisfied? Proof
subspace-hereditary property of topological spaces Yes functional Hausdorffness is hereditary
refining-preserved property of topological spaces Yes functional Hausdorffness is refining-preserved
product-closed property of topological spaces Yes functional Hausdorffness is product-closed
box product-closed property of topological spaces Yes functional Hausdorffness is box product-closed