Let $i:X\rightarrow \mathbb{R}^N$ be an imbedding of a topological space $X$. Assume that there exists an open neighborhood $U$ containing $i(X)$ which also admits a retraction $p:U\rightarrow X$. The question is whether any two such retractions $p\_1$ and $p\_2$ are homotopic? (It is allowed to shrink $U$ is needed.)



Retractions of ENR
Yes. Consider $H: U \times [0; 1] \to \mathbb{R}^N$, $H\_t=tp\_1+(1-t)p\_2$. Clearly, $H^{-1}(U)$ is an open neighbourhood of $\iota(X) \times [0, 1]$, by compactness of $[0; 1]$ it contains an open set of of the form $\tilde{U} \times [0, 1]$ with $\tilde{U}$ open neighbourhood of $\iota(X)$, so for example $p\_1 \circ H |\_{\tilde{U}}$ is a homotopy between the two retracts on $\tilde{U}$.


