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Brouwer's theorem of invariance of dimension under homeomorphism holds for non-empty open subsets of $\mathbb{R}^{n}$:

Let $\emptyset\neq U$ open $\subseteq\mathbb{R}^{n}$ and $\emptyset\neq V$ open $\subseteq\mathbb{R}^{m}$. If $U$ and $V$ are homeomorphic, then $n=m$.

A domain is any non-empty connected open set in a topological space. The closure of a domain is called a closed domain. (Encyclopedia of Mathematics: Domain)

Does the following theorem hold also:

Let $\emptyset\neq U \subset \mathbb{R}^{n}$ be a closed domain in $\mathbb{R}^{n}$, and let $\emptyset\neq V \subset \mathbb{R}^{m}$ be a closed domain in $\mathbb{R}^{m}$. If $U$ and $V$ are homeomorphic, then $n=m$.

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    $\begingroup$ Any homeomorphism $f$ from $U$ to $V$ restricts to a homeomorphism from $U'$ to $f(U')$, for $U$ a nonempty open subset of $U$. Such a $U'$ exists since $U$ is a closed domain (e.g. take some open domain whose closure is $U$). Now apply invariance of domain. $\endgroup$ Jun 25 '17 at 20:12
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    $\begingroup$ (By the way, note that the converse fails: a homeomorphism between two opens does not extend to a homeomorphism between their closures, think of $(0, 1)$ versus the circle minus a point.) $\endgroup$ Jun 25 '17 at 20:13
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I now found the following answer.

The supposed theorem follows also from the following theorem of Emanual Sperner proved in the article below:

"A point set of $R_{n+h}$ ($h>0$) containing inner points cannot be homeomorphic to any point set of $R_{n}$."

It follows from the following theorem of Lebesgue:

A bounded point set $G$ in the $n$-dimensional number space is given, which contains inner points. Given a bounded point set $G$ in the $n$-dimensional number space containing inner points. The points of $G$ be distributed to a finite number of closed sets $M_{i}$ ($i=1,2,3,...,s$), so that every point of $G$ occurs at least in one of the sets $M_{i}$. Then there is at least one point in $G$ which lies in at least $n+1$ sets, if only the $M_{i}$ were chosen sufficiently small.

Sperner, Emanuel: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. In: Abh. Math. Sem. Univ. Hamburg. Band 6, 1928, 265-272

Others reported the following simple proof of the supposed theorem:

de.sci.mathematik: Invarianz der Dimension auch für abgeschlossene Gebiete gültig? 11. Juli (in German) (Google Groups)

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