open map

This page is about the concept in topology. For the more general concept see at open morphism.



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory




(open maps and closed maps)

A continuous function f:(X,τ X)(Y,τ Y)f \colon (X,\tau_X) \to (Y, \tau_Y) between topological spaces is called



(image projections of open/closed maps are themselves open/closed)

If a continuous function f:(X,τ X)(Y,τ Y)f \colon (X,\tau_X) \to (Y,\tau_Y) is an open map or closed map (def. 1) then so is its image projection Xf(X)YX \to f(X) \subset Y, respectively, for f(X)Yf(X) \subset Y regarded with its subspace topology.


If ff is an open map, and OXO \subset X is an open subset, so that f(O)Yf(O) \subset Y is also open in YY, then, since f(O)=f(O)f(X)f(O) = f(O) \cap f(X), it is also still open in the subspace topology, hence Xf(X)X \to f(X) is an open map.

If ff is a closed map, and CXC \subset X is a closed subset so that also f(C)Yf(C) \subset Y is a closed subset, then the complement Y\f(C)Y \backslash f(C) is open in YY and hence (Y\f(C))f(X)=f(X)\f(C)(Y \backslash f(C)) \cap f(X) = f(X) \backslash f(C) is open in the subspace topology, which means that f(C)f(C) is closed in the subspace topology.


(projections out of product spaces are open maps)

For (X 1,τ X 1)(X_1,\tau_{X_1}) and (X 2,τ X 2)(X_2,\tau_{X_2}) two topological spaces, then the projection maps

pr i:(X 1×X 2,τ X 1×X 2)(X i,τ X i) pr_i \;\colon\; (X_1 \times X_2, \tau_{X_1 \times X_2}) \longrightarrow (X_i, \tau_{X_i})

out of their product topological space

X 1×X 2 pr 1 X 1 (x 1,x 2) AAA x 1 \array{ X_1 \times X_2 &\overset{pr_1}{\longrightarrow}& X_1 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_1 }
X 1×X 2 pr 2 X 2 (x 1,x 2) AAA x 2 \array{ X_1 \times X_2 &\overset{pr_2}{\longrightarrow}& X_2 \\ (x_1, x_2) &\overset{\phantom{AAA}}{\mapsto}& x_2 }

are open continuous functions (def. 1).

This is because, by definition, every open subset OX 1×X 2O \subset X_1 \times X_2 in the product space topology is a union of products of open subsets U iX 1U_i \in X_1 and V iX 2V_i \in X_2 in the factor spaces

O=iI(U i×V i) O = \underset{i \in I}{\cup} \left( U_i \times V_i \right)

and because taking the image of a function preserves unions of subsets

pr 1(iI(U i×V i)) =iIpr 1(U i×V i) =iIU i. \begin{aligned} pr_1\left( \underset{i \in I}{\cup} \left( U_i \times V_i \right) \right) & = \underset{i \in I}{\cup} pr_1 \left( U_i \times V_i \right) \\ & = \underset{i \in I}{\cup} U_i \end{aligned} \,.

Revised on June 7, 2017 07:34:53 by Urs Schreiber (