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Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$,

$H^{i}(X-A;\mathbb Z)\cong H^{i}(X-N;\mathbb Z)$

My guess: if two subspaces $A$ and $B$ of $X$ are homotopy equivalent then their complements $X-A$ and $X-B$ must be homotopy equivalent and then have the same cohomology groups?

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    Is the following true: if $N$ deformation retracts onto $A$ then $X-A$ deformation retracts onto $X-N$?2012-10-11

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