Final Topology - Examples

Examples

  • The quotient topology is the final topology on the quotient space with respect to the quotient map.
  • The disjoint union is the final topology with respect to the family of canonical injections.
  • More generally, a topological space is coherent with a family of subspaces if it has the final topology coinduced by the inclusion maps.
  • The direct limit of any direct system of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms.
  • Given a family of topologies {τi} on a fixed set X the final topology on X with respect to the functions idX : (X, τi) → X is the infimum (or meet) of the topologies {τi} in the lattice of topologies on X. That is, the final topology τ is the intersection of the topologies {τi}.

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