Topological spaces

Abbreviation: Top

Definition

A topological space is a structure X = (X,τ), where τ = Ω(X) ⊆ P(X) is a collection of subsets of X called the open sets of X such that
any union of open sets is open:   U ⊆ Ω(X)  ⇒  ∪U ∈ Ω(X) and
any finite intersection of open sets is open:   U,V ∈ Ω(X)  ⇒  U∩V ∈ Ω(X) and X ∈ Ω(X).

Remark: Note that the union of an empty collection is empty, so Ø ∈ Ω(X).
The set of closed sets of X is K(X) = {X−U | U ∈ Ω(X)}.

Morphisms

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

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