Separation Axiom - Main Definitions

Main Definitions

These definitions all use essentially the preliminary definitions above.

Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T₄" are sometimes interchanged, similarly "regular" and "T₃", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.

Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.

In all of the following definitions, X is again a topological space, and all functions are supposed to be continuous.

• X is T₀, or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It will be a common theme among the separation axioms to have one version of an axiom that requires T₀ and one version that doesn't.)

• X is R₀, or symmetric, if any two topologically distinguishable points in X are separated.

• X is T₁, or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T₁ if and only if it is both T₀ and R₀. (Although you may say such things as "T₁ space", "Fréchet topology", and "Suppose that the topological space X is Fréchet", avoid saying "Fréchet space" in this context, since there is another entirely different notion of Fréchet space in functional analysis.)

• X is R₁, or preregular, if any two topologically distinguishable points in X are separated by neighbourhoods. An R₁ space must also be R₀.

• X is Hausdorff, or T₂ or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T₀ and R₁. A Hausdorff space must also be T₁.

• X is T_2½, or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T_2½ space must also be Hausdorff.

• X is completely Hausdorff, or completely T₂, if any two distinct points in X are separated by a function. A completely Hausdorff space must also be T_2½.

• X is regular if, given any point x and closed set F in X, x does not belong to F, then they are separated by neighbourhoods. (In fact, in a regular space, any such x and F will also be separated by closed neighbourhoods.) A regular space must also be R₁.

• X is regular Hausdorff, or T₃, if it is both T₀ and regular. A regular Hausdorff space must also be T_2½.

• X is completely regular if, given any point x and closed set F in X, x does not belong to F, then they are separated by a function. A completely regular space must also be regular.

• X is Tychonoff, or T_3½, completely T₃, or completely regular Hausdorff, if it is both T₀ and completely regular. A Tychonoff space must also be both regular Hausdorff and completely Hausdorff.

• X is normal if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, in a normal space, any two disjoint closed sets will also be separated by a function; this is Urysohn's lemma.)

• X is normal Hausdorff, or T₄, if it is both T₁ and normal. A normal Hausdorff space must also be both Tychonoff and normal regular.

• X is completely normal if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.

• X is completely normal Hausdorff, or T₅ or completely T₄, if it is both completely normal and T₁. A completely normal Hausdorff space must also be normal Hausdorff.

• X is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal.

• X is perfectly normal Hausdorff, or T₆ or perfectly T₄, if it is both perfectly normal and T₁. A perfectly normal Hausdorff space must also be completely normal Hausdorff.