locally proper map




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



The relative version of a locally compact topological space is a continuous map that behaves like a ‘family of locally compact spaces’.



A continuous function f:XYf \colon X \to Y is called locally proper if for any xXx \in X and neighbourhood VxV \ni x there is a neighbourhood AxA\ni x with AVA \subset V and a neighbourhood Uf(x)U \ni f(x) such that AUA \to U is a proper map.


  • The inclusion map of a locally closed subset CXC\hookrightarrow X in a topological space XX is locally proper.

  • A topological space XX is a locally compact space if and only if X*X\to \ast is locally proper.



Let f:XYf\colon X\to Y and g:YZg\colon Y\to Z be locally proper. Then gfg\circ f is locally proper. If WYW\to Y is any continuous function then W× YZWW\times_Y Z \to W is locally proper. (ie locally proper maps are closed under composition and stable under pullback. Hence they form a singleton coverage)

As a corollary, one has that for a locally compact space XX, every locally closed subspace of XX is locally compact.


Let f:XYf\colon X\to Y be a continuous function. Then if

  • there is an open covering 𝒰={U i}\mathcal{U} = \{U_i\} of XX such that f| U i:U iYf\big|_{U_i} \colon U_i \to Y is locally proper for all ii,


  • there is an open covering 𝒱={V j}\mathcal{V} = \{V_j\} of YY such that f 1(V j)V jf^{-1}(V_j) \to V_j is locally proper for all jj,

then ff is locally proper. The converse implications also hold

Recall that a continuous map g:YZg\colon Y \to Z of topological spaces is called separated if YY× ZYY\to Y\times_Z Y is a closed embedding.


If f:XYf\colon X\to Y and g:YZg\colon Y\to Z are continuous maps, gg is separated and gfg\circ f is locally proper, then ff is locally proper


Every continuous map from a locally compact Hausdorff space to a Hausdorff space is separated and locally proper.

The following proposition generalises the well-known result that compact Hausdorff spaces are locally compact.


Every separated and proper map is locally proper

The proper base change theorem also holds for locally proper maps if one uses direct image with compact support instead of ordinary direct image


  • Olaf M. Schnürer and Wolfgang Soergel, Proper base change for separated locally proper maps, Rend. Sem. Mat. Univ. Padova 135 (2016) pp 223-250. doi:10.4171/RSMUP/135-13, arXiv:1404.7630
  • Günther Richter and Alexander Vauth, Fibrewise sobriety, in: Categorical structures and their applications. Proceedings of the North-West European category seminar, Berlin, Germany, March 28–29, 2003. River Edge, NJ: World Scientific. 265–283 (2004; Zbl 1065.18003). doi:10.1142/9789812702418_0020

Last revised on February 5, 2020 at 11:39:19. See the history of this page for a list of all contributions to it.