Gol D Roger

I've been searching in Google for a while but can't find anything related to this question:
If X is a locally compact space and U is an open subset of X, show that U is locally compact.

This subspace:


and well, if $Y$ is a subspace of $Z$, we say that $Y$ is a retract of $Z$ if there is a continuous map $r: Z \to Y$ such that $r(y) = y$ for each $y \in Y$.

Ok, Munkres gave me this one:

Let $Y$ be a normal space. Then $Y$ is said to be an absolute retract if for every pair of spaces $(Y_0 , Z)$ such that $Z$ is normal and $Y_0$ is a closed subspace of $Z$ homeomorphic to $Y$, the space $Y_0$ is a retract of $Z$.

Fourier Analysis and Partial Differential Equations

by Rafael Iorio Jr

I need to find an homeomorphism

$
\begin{align*}
f: U_0 \to V_1,
\end{align*}
$

(where $U_0$ is an open neighborhood of the identity element $0$ in $(\mathbb{R}, +)$, and $V_1}$ is an open neighborhood of the identity element $1$ in $(\mathbb{S}^1 , \cdot )$),

such that:

$f(x+y) = f(x) \cdot f(y), \ \forall x,y \in U_0$ such that $x+y \in U_0$,

but the following condition doesn't follows:

$f^{-1}(z \cdot t) = f{-1}(z)+f{-1}(t), \ \forall z,t \in V_1$ such that $z \cdot t \in V_1$.