# Topological spaces

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Changed: 6,10c6,12
 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).
 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)}.

Changed: 16c18
 V ∈ Ω(Y)  ⇒  f−1[V] ∈ Ω(X).
 V ∈ Ω(Y)  ⇒  f−1[V] ∈ Ω(X).

Changed: 29,30c31

 Classtype second-order

 Classtype second-order

Changed: 33,34c34

 Amalgamation property yes

 Amalgamation property yes

Changed: 37,38c37

 Strong amalgamation property no

 Strong amalgamation property yes

Changed: 41,42c40

 Epimorphisms are surjective no

 Epimorphisms are surjective yes

Added: 44a43,46
 Remark: The properties given above use an (E,M) factorization system with E =  surjective morphisms and M =  embeddings.

Changed: 47c49
 [T 0-spaces]?
 T0-spaces

Changed: 51c53
 ...?
 Sets

Removed: 53d54

Changed: 55c56,59
 http://www.wikipedia.org/wiki/Topological_space

### References

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

### 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)  ⇒  UV ∈ 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) = {XU | U ∈ Ω(X)}.

### Morphisms

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

### 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.

T0-spaces

Sets

### References

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Last edited August 22, 2003 9:34 pm (diff)
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