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A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and $\pi_2|U \text{rel } \vartriangle_X$, where $\pi_1, \pi_2$ are the projections onto the first and the second factor.

FACT: If $X$ is perfectly normal Hausdorff space with a perfectly normal square then $X$ is ULC if and only if the diagonal embedding $\vartriangle_X \to X \times X$ is a cofibration.

I found the following fact (without proof) in Warner's book Topic in Topology and Homotopy Theory and have been unable to prove it myself:

Let $i: A \to X$ be a closed cofibration, $f: A \to Y$ a continuous map and $X, Y$ ULC perfectly normal Hausdorff spaces with perfectly normal squares. Then the adjunction space $Z=X \cup_f Y$ is ULC provided its square is perfectly normal.

Any suggestions on how to prove this will be appreciated. Thanks!

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