page_title: Separation axiom
hierarchical_section_title: Separation axiom
caption_attribution_description: English: Simple illustration of Hausdorff, Regular, and Normal spaces for use as visual aid to the accompanying article on the separation axioms.
context_page_description: In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
  The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces.
  However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom, which means "separation axiom."
  The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms. It is important to understand the authors' definition of each condition mentioned to know exactly what they mean, especially when reading older literature.
context_section_description: In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff.
  The separation axioms are axioms only in the sense that, when defining the notion of topological space, one could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces.
  However, the term "separation axiom" has stuck. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom, which means "separation axiom."
  The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms. It is important to understand the authors' definition of each condition mentioned to know exactly what they mean, especially when reading older literature.
caption: An illustration of some of the separation axioms. Grey amorphous broken-outline regions indicate open sets surrounding disjoint closed sets or points: red solid-outline circles denote closed sets while black dots represent points.
exif: {}