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Aim of this question is to clarify the differences between the two concepts written in the title, because it's unclear to me wether one condition implies the other, under which circumstances they are equivalent or not. More importantly, I didn't find any reference for this comparison, so I'm asking for your help.

We begin with the definition, taken from [tom Dieck's Algebraic Topology]:

Definition 1 [Deformation Retract] A subspace $K \subset X$ is a deformation retract iff $\exists$ an homotopy relative to $K$ $h_t \colon X \to X$ s.t. $h_0=Id_X$ and $h_1 = i\circ r$, where $i$ is the inclusion $K\subset X$ and $r\colon X \to K$ is the so-called retraction (i.e $r\circ i = Id_K$)

Definition 2 [ENR] $K$ is a ENR if $\exists \ j \colon K \to \mathbb{R}^n$, an embedding, and exists an open nbd $U$ of $K':=j(K)$, and a retraction $r\colon U \to K$ s.t. $j\circ r=Id_{K'}$

To be precise, on [tom Dieck] def. $2$ is stated as above with the only difference that the retraction is required to satisfy $j\circ r = Id(K)$ (page 448) but I don't know what does it mean $Id(K)$ and the domain of $r$ is contained in $\mathbb{R}^n$, so cannot be $K$.

First observation, if $K$ is a subset of some $\mathbb{R}^n$, then setting $j$ as the inclusion, we retrieve the definition of a retract, i.e exists a retraction which leaves $K=K'$ untouched.

And now the my reasoning.

According to my understandings, any retract $K$ of an open subset of $\mathbb{R}^n$ seems to be an ENR according to def. 2. Then noticing that any open subset of $\mathbb{R}^n$ is trivially ENR, we can apply the following result [tom Dieck Proposition 18.4.3]

Proposition 18.4.3 Let $X$ be an ENR. Suppose $f_0,f_1 \colon Y \to X$ are maps which coincide on a subset $B\subset Y$. Then there exists a neighbourhood $W$ of $B$ in $Y$ and a homotopy $h \colon f_0|W \simeq_{rel B} \ \ f_1|W$

Take $Y=W=X$ and consider $f_0=Id_W$ and $f_1=i\circ r$ where $W \subset \mathbb{R}^n$ is an open nbd of $K$ ENR. $W$ is ENR and using the prop ($B$ is $K$ clearly) we have that $K$ is a deformation retract of $W'\subset W$.

So in conclusion, it seems to be true that any retract of an open subset of $\mathbb{R}^n$ is a deformation retract of some open nbd, and up to restriction, any retract of some opens, is instead a def. retract of it.

So to recap, I drew these conclusions:

1) In the case $X \subset \mathbb{R}^n$, ENR if an only if Deformation retract (up to restriction of the open nbd)

2) In the general case, if the nbd of $X$ is ENR on its own, then (up to restriction)the two notions are equivalent.

Is this correct?

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