Coherent Topology - Properties

Properties

Let X be coherent with a family of subspaces {C_α}. 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 = {C_α}. 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 C_α is determined by the family of all D_β contained in C_α then X is determined by D.

Let X be determined by {C_α} and let Y be an open or closed subspace of X. Then Y is determined by {Y ∩ C_α}.

Let X be determined by {C_α} and let f : X → Y be a quotient map. Then Y is determined by {f(C_α)}.

Let f : X → Y be a surjective map and suppose Y is determined by {D_α : α ∈ A}. For each α ∈ A let

be the restriction of f to f−1(D_α). Then

• If f is continuous and each f_α is a quotient map, then f is a quotient map.
• f is a closed map (resp. open map) if and only if each f_α is closed (resp. open).