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The name separated space is also used. A related, but weaker, notion is that of a preregular space. is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called an R 1 space. The relationship between these two conditions is as follows.
As an example we will define the T 2 axiom, which is the condition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods. Separated spaces are usually called Hausdorff spaces or T 2 spaces.
In topology, a discipline within mathematics, an Urysohn space, or T 2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous function.
A T 5 space, or completely T 4 space, is a completely normal T 1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T 4 space. A perfectly normal space is a topological space in which every two disjoint closed sets and can be precisely separated by a function, in the sense that there is a continuous function ...
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. T 6 or Perfectly normal Hausdorff, or perfectly T 4. A space is perfectly normal Hausdorff, if it is both perfectly normal and T 1. A perfectly normal Hausdorff space must also be completely ...
A subspace of a separable space need not be separable (see the Sorgenfrey plane and the Moore plane), but every open subspace of a separable space is separable (Willard 1970, Th 16.4b). Also every subspace of a separable metric space is separable. In fact, every topological space is a subspace of a separable space of the same cardinality.
Every subspace of a G δ space is a G δ space. Every metrizable space is a G δ space. The same holds for pseudometrizable spaces. Every second countable regular space is a G δ space. This follows from the Urysohn's metrization theorem in the Hausdorff case, but can easily be shown directly. [3] Every countable regular space is a G δ space.
A topological space in which every two disjoint closed subsets and are precisely separated by a continuous function is perfectly normal. Urysohn's lemma has led to the formulation of other topological properties such as the 'Tychonoff property' and 'completely Hausdorff spaces'.