By Satz 1 of Dold's Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft, it suffices that there exist a continuous function $\tau \colon B \to [0,1]$ with $(\overline{A} \cap V) \subset \tau^{-1}((0,1]) \subset V$. Interestingly, there is also a converse, Satz 2: if $\{V_i\}_{i \in I}$ is a numerable open cover $X$ such that each $A \cap V_i \hookrightarrow V_i$ is Hurewicz cofibration, then $A \hookrightarrow X$ is Hurewicz cofibration.

By Satz 1 of Dold's Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft, it suffices that there exist a continuous function $\tau \colon X \to [0,1]$ with $(\overline{A} \cap V) \subset \tau^{-1}((0,1]) \subset V$. Interestingly, there is also a converse, Satz 2: if $\{V_i\}_{i \in I}$ is a numerable open cover $X$ such that each $A \cap V_i \hookrightarrow V_i$ is Hurewicz cofibration, then $A \hookrightarrow X$ is Hurewicz cofibration.