Are Sperner's Lemma and Brouwer's Fixed Point Theorem equivalent? On this Wikipedia link I found the following statement:

In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.

At first, I found this statement very strange. How can two theorems be equivalent? Of course you could say that
$$``Sperner's~lemma~is~true"~\iff~``Brouwer~fixed~point~theorem~is~true"$$
since both statements are, in fact, true, but you could say the same of any two theorems proved true, and I think I would be considered a fool to say that Fermat's little theorem is equivalent to Pythagoras' theorem.
The thing is, it makes sense to talk about property equivalence, and I found that Brouwer's fixed point theorem is, in fact, related to a property of topological spaces.

A topological space $(X, \tau)$ is called a fixed-point domain if any continuous function $f:X\rightarrow X$ has a fixed point.

Later I realized that "equivalent" probably just referred to having a demonstration of one passing through the other and vice versa, but my confusion motivated a strong curiosity: Is there any property about topological spaces that is equivalent to "being a fixed-point domain" and somehow generalizes Sperner's lemma? I know that "triangulation" is a well-defined concept in topology, which leads me to believe that the answer to my question might be yes.
An answer to the case where $X$ is a subspace of $\Bbb{R}^n$ would already satisfy me a lot.
 A: Very nice question! You are right, any two true statements are equivalent, but I think the claim that Sperner's lemma is equivalent to the Brouwer fixed point theorem should not be understood on that level.
The "standard" proof of the Brouwer fixed point theorem in arbitray dimension $n$ is based on the machinery of algebraic topology; it involves the construction of homology groups of topological spaces (plus proving their basic properties) and the calculation of the homology groups of spheres.
Sperner's lemma is not (at least in my opinon) a combinatorial analog of the Brouwer fixed point theorem, but a combinatorial theorem dealing with certain properties of simplicial subdivisions of a standard simplex.
Using Sperner's lemma one can easily prove the Brouwer fixed point theorem (see here), but I do not think that there is a simple derivation of Sperner's lemma from the Brouwer fixed point theorem. In fact, the usual proof of Sperner's lemma is fairly elementary and has nothing to do with topology. This does not exclude that there is a nice way to invoke the Brouwer fixed point theorem, but it seems inadequate to replace elementary combinatorial arguments by sophisticated topological arguments.
