every two distinct points have disjoint neighbourhoods. T2 spaces are always T1. T2½ or Urysohn. A space is Urysohn if every two distinct points have disjoint closed neighbourhoods. T2½ spaces are always T2. Completely T2 or completely Hausdorff. A space is completely T2 if every two distinct points are separated by a function. Every completely Hausdorff space is Urysohn. Regular. A space is regular if whenever C is a closed set and p is a point not in C, then C and p have disjoint neighbourhoods. T3 or Regular Hausdorff. A space is regular Hausdorff if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Completely regular. A space is completely regular if whenever C is a closed set and p is a point not in C, then C and {p} are separated by a function. T3½, Tychonoff, Completely regular Hausdorff or Completely T3. A Tychonoff space is a completely regular T0 space. (A completely regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) Tychonoff spaces are always regular Hausdorff. Normal. A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Normal spaces admit partitions of unity. T4 or Normal Hausdorff. A normal space is Hausdorff if and only if it is T1. Normal Hausdorff spaces are always Tychonoff. Completely normal. A space is completely normal if any two separated sets have disjoint neighbourhoods. T5 or Completely normal Hausdorff. A completely normal space is Hausdorff if and only if it is T1. Completely normal Hausdorff spaces are always normal Hausdorff. Perfectly normal. A space is perfectly normal if any two disjoint closed sets are precisely separated by a function. A perfectly normal space must also be completely normal. T6 or Perfectly normal Hausdorff, or perfectly T4. A space is perfectly normal Hausdorff, if it is both perfectly normal and T1. A perfectly normal Hausdorff space must also be completely normal Hausdorff. Discrete space. A space is discrete if all of its points are completely isolated, i.e. if any subset is open. Number of isolated points. The number of isolated points of a topological space. Countability conditions Separable. A space is separable if it has a countable dense subset. First-countable. A space is first-countable if every point has a countable local base. Second-countable. A space is second-countable