−Table of Contents
Topological spaces
Abbreviation: Top
Definition
A \emph{topological space} is a structure X=⟨X,τ⟩, where τ=Ω(X)⊆P(X) is a collection of subsets of X called the \emph{open sets of} X such that
any union of open sets is open: U⊆Ω(X)⟹⋃U∈Ω(X)
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 ∅∈Ω(X).
The set of \emph{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 \emph{continuous}:
V∈Ω(Y)⟹f−1[V]∈Ω(X)
Examples
Example 1:
Basic results
Properties
| Classtype | second-order |
|---|---|
| Amalgamation property | yes |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | yes |
Remark: The properties given above use an (E,M) factorization system with E= surjective morphisms and M= embeddings.
