Example

Let B be a topological space and let $(A_i)_{i\in I}$ be a covering of B; let A denote the sum space of the family $(A_i)_{i\in I}$, and let $p: A\rightarrow B$ be the canonical mapping. The mapping $p$ is universally strict under each of the following two assumptions:

(i) For every point $b\in B$, there exists $i\in I$ such that $b$ is an interior point of $A_i$;

(ii) The family $(A_i)$ is locally finite and the $A_i$ are closed subsets of B.

Under assumption (i), the mapping $p$ possesses in a neighbourhood of every point a continuous section. Under assumption (ii), the mapping $p$ is proper by definition of the sum space (resp. from TG, I, p. 6, prop. 4 and TG, I, p. 75, th. 1). It is therefore universally strict.