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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.