T 1-space

attainable space

A topological space $ X $ in which the closure of any one-point set coincides with itself. This is equivalent to the requirement that the intersection of all neighbourhoods of a point $ x \in X $ is identical with $ x $ or that, for any two different points $ x , y \in X $, there exist neighbourhoods $ U _ {x} $ and $ U _ {y} $ of them such that $ U _ {x} \Nso y $ and $ U _ {y} \Nso x $, i.e. that the separation axiom $ T _ {1} $ holds.

Attainability, i.e. the property that $ T _ {1} $ holds, is a hereditary property: Any subspace of a $ T _ {1} $- space is a $ T _ {1} $- space, and a topology majorizing the topology of a $ T _ {1} $- space is a $ T _ {1} $- topology. Any $ T _ {2} $- space (cf. Hausdorff space) is $ T _ {1} $- space, but the converse is not true: There exist $ T _ {1} $- spaces which are not $ T _ {2} $- spaces. These include, for example, an infinite set $ \beta $ with the topology in which the sets with finite complements are considered to be open.

Comments

A very important class of spaces that are $ T _ {1} $ but, as a rule, not $ T _ {2} $ are the spectra $ \mathop{\rm Spec} ( A) $ of rings $ A $ with the Zariski topology, cf. Affine scheme.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)