composition with coercive function

Theorem 1.

Suppose X,Y,Z are topological spacesMathworldPlanetmath, f:XY is a bijectiveMathworldPlanetmathPlanetmath proper map, and g:YZ is a coercive map. Then gf:XZ is a coercive map.


Let JZ be a compact set. As g is coercive, there is a compact set KY such that


Let I=f-1(K), and since f is a proper map I is compact. Thus


and gf is coercive. ∎

Title composition with coercive function
Canonical name CompositionWithCoerciveFunction
Date of creation 2013-03-22 15:20:16
Last modified on 2013-03-22 15:20:16
Owner matte (1858)
Last modified by matte (1858)
Numerical id 6
Author matte (1858)
Entry type Theorem
Classification msc 54A05