T_1 Spaces: definition and relation to T_0 and T_2 (Hausdorff) Spaces

Описание к видео T_1 Spaces: definition and relation to T_0 and T_2 (Hausdorff) Spaces

The ideas of T_0, T_1, T_2 spaces describe a progressively more “expensive” way to separate points in the space. The T_1 property ensures any two distinct points in the space have neighborhoods that miss the other point. This is perhaps requires more open sets than the T_0 property (more expensive). But we don’t require that we can always find disjoint neighborhoods that separate the points. So T_2 perhaps requires more open sets than T_1 (more expensive). From the definitions one can see T_2 spaces are T_1 and T_0, and that T_1 spaces are T_0. We look at examples to show the converse of these statements may be false, meaning we consider a T_0 space that is not T_1, and a T_1 space that is not T_2.

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