Properties
Let X be coherent with a family of subspaces {Cα}. A map f : X → Y is continuous if and only if the restrictions
are continuous for each α ∈ A. This universal property characterizes coherent topologies in the sense that a space X is coherent with C if and only if this property holds for all spaces Y and all functions f : X → Y.
Let X be determined by a cover C = {Cα}. Then
- If C is a refinement of a cover D, then X is determined by D.
- If D is a refinement of C and each Cα is determined by the family of all Dβ contained in Cα then X is determined by D.
Let X be determined by {Cα} and let Y be an open or closed subspace of X. Then Y is determined by {Y ∩ Cα}.
Let X be determined by {Cα} and let f : X → Y be a quotient map. Then Y is determined by {f(Cα)}.
Let f : X → Y be a surjective map and suppose Y is determined by {Dα : α ∈ A}. For each α ∈ A let
be the restriction of f to f−1(Dα). Then
- If f is continuous and each fα is a quotient map, then f is a quotient map.
- f is a closed map (resp. open map) if and only if each fα is closed (resp. open).
Read more about this topic: Coherent Topology
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