At wikipedia article about Brouwer fixed-point theorem, in the second paragraph, one can read the following:
In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.
My question is: in what sense does these four theorems characterize the Euclidean space? Is there a theorem that says: "If a space $X$ fulfils the criteria of those four theorems, then $X\simeq A\subseteq \mathbb{R}^n$ " or something along the lines?
Wikipedia has a citation at that point, namely "See page 15 of: D. Leborgne Calcul différentiel et géométrie". Sadly, I don't speak French. An English source or a translation would be apprieciated.
Considering that the hairy ball theorem and the Borsuk–Ulam theorem are theorems about spheres, not $\mathbb{R}^n$, I cannot see how they can characterize $\mathbb{R}^n$ in a meaningful way.
I do not have access to Leborgne's book, but considering the quote is apparently from page 15, I would assume that this is from the introduction, and that "characterize" is not used in a formal sense.
Rather, I would say that these theorems are nice truths about Euclidean spaces (or spheres...) that were historically important (and remain so). For example the Jordan curve theorem is an intuitive truth about $\mathbb{R}^n$, and the fact that it is so hard to prove is rather crazy and shattered many people's expectations.