T1-spaces

Abbreviation: Top₁

Definition

A T₁-space is a topological space X = (X,Ω(X)) such that
for every pair of distinct points in the space, there is a pair of open sets containing each point but not the other:   x,y ∈ X  ⇒  ∃U,V ∈ Ω(X)[x ∈ U \ V and y ∈ V \ U].

Morphisms

Let X and Y be T₁-spaces. A morphism from X to Y is a function f : X→Y that is continuous: V ∈ Ω(Y)  ⇒  f⁻¹[V] ∈ Ω(X).

Definition

A T₁-space is a topological space X = (X,Ω(X)) such that all singleton subsets are closed:   X \ {x} ∈ Ω(X).

Some results

Examples

Properties

Remark: The properties given above use an (E,M) factorization system with E =  surjective morphisms and M =  embeddings.

Subclasses

Superclasses

References

http://www.wikipedia.org/wiki/Topology_glossary