General topology/Constructions

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Lattice of topologies on a fixed set[edit]

Proposition (topologies on a fixed set are a complete lattice):

Let be a set, and let be a family of topologies on . Then there exists a

Final and initial topologies[edit]

Proposition (preimage of a topology is a topology):

Let be a function from a set to another set , and let be a topology on . Then is a topology on .

Proof: This follows instantly from the formulae

  1. ,
  2. and for any family of subsets of

Definition (initial topology):

Let be a set and let be a family of topological spaces. Let further () be functions. The initial topology on