Contents

# Contents

## Idea

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

## Definition

###### Definition

A continuous function $f \colon X \to Y$ is called locally proper if for any $x \in X$ and neighbourhood $V \ni x$ there is a neighbourhood $A\ni x$ with $A \subset V$ and a neighbourhood $U \ni f(x)$ such that $A \to U$ is a proper map.

## Examples

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

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

## Properties

###### Proposition

Let $f\colon X\to Y$ and $g\colon Y\to Z$ be locally proper. Then $g\circ f$ is locally proper. If $W\to Y$ is any continuous function then $W\times_Y Z \to W$ 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 $X$, every locally closed subspace of $X$ is locally compact.

###### Proposition

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

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

or

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

then $f$ is locally proper. The converse implications also hold

Recall that a continuous map $g\colon Y \to Z$ of topological spaces is called separated if $Y\to Y\times_Z Y$ is a closed embedding.

###### Proposition

If $f\colon X\to Y$ and $g\colon Y\to Z$ are continuous maps, $g$ is separated and $g\circ f$ is locally proper, then $f$ is locally proper

###### Corollary

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.

###### Proposition

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