I've asked this on MathSE without success:


A pair of topological spaces $(X,A)$ is an NDR (neighborhood deformation retract) pair if there are continuous maps $u:X\to I$ and $h:X\times I\to X$ satisfying, (1) $u^{-1}(0)=A$, (2) For all $x\in X$, $h(x,0)=x$, (3) For all $a\in A$ and $t\in I$, $h(a,t)=a$, (4) If $u(x)<1$, then $h(x,1)\in A$.

By exercise I.E.6 in Spanier, a theorem on p. 43 of May's Concise Course, or Thm. 7.1.10 in Selick's Intro To Homotopy Theory, $(X,A)$ is an NDR pair if and only if $A\to X$ is a cofibration.

My question is: is being an NDR pair a local property? That is, if there is an open cover $\{U_i\}$ of $X$ with each pair $(U_i,A\cap U_i)$ an NDR pair, can we conclude that $(X,A)$ is an NDR pair?

I think this can be shown to hold if the open cover is finite and $X$ satisfies some separation axioms by using a partition of unity together with the deformation retraction appearing in the NDR definition to show the homotopy extension property defining cofibration holds (although I might be too optimistic). A reference in the literature to this result would be ideal although I would also be grateful for any counterexamples or explanations of why it isn't true.

I've looked through the above mentioned references and skimmed through Hatcher but haven't been able to find it although it might follow from certain results on pushouts of cofibrations. I haven't been able to get my hands on Whitehead's big book yet.


There is a theorem of Dold to this effect:

Dold, Albrecht Die Homotopieerweiterungseigenschaft (=HEP) ist eine lokale Eigenschaft. (German) Invent. Math. 6 1968 185–189.

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