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Topological spacesTopological spaces|
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 \mathbf{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 ∈ τ and X ∈ Ω(X). Remark: Note that the union of an empty collection is empty, so Ø ∈ Ω(X). |
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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). |
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V ∈ Ω(Y) ⇒ f−1[V] ∈ Ω(X). |
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V ∈ Ω(Y) ⇒ f−1[V] ∈ Ω(X). |
| Classtype | second-order |
| Classtype | second-order |
| Amalgamation property | yes |
| Amalgamation property | yes |
| Strong amalgamation property | no |
| Strong amalgamation property | yes |
| Epimorphisms are surjective | no |
| Epimorphisms are surjective | yes |
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Remark: |
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[T 0-spaces]? |
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T0-spaces |
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...? |
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Sets |
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http://www.wikipedia.org/wiki/Topological_space |
Referenceshttp://www.wikipedia.org/wiki/Topological_space |
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)}.
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).
| 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.
http://www.wikipedia.org/wiki/Topological_space