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Knaster-Kuratowski-Mazurkiewicz lemma

Triangle KKM lemma

Let $\Delta^n$ be an $n$-dimensional <simplex>, $n\geq1$, with <barycentric coordinates> $x_0,\ldots,x_n$. Suppose closed <sets> $F_0,\ldots,F_n$ cover it and each $F_i$ contains the opposite <face of a simplex> $\{x_i=0\}$. Then their common <intersection> is nonempty. This is the higher-dimensional opposite-face version of the <triangle KKM lemma>.

If the common <intersection> were empty, normalized <distance functions> to the $F_i$ would give a <continuous map> $g:\Delta^n\to\partial\Delta^n$ carrying every <face of a simplex> into itself. On the <boundary>, the straight interpolation $(1-t)x+tg(x)$ stays in the same <face of a simplex>, so $g|_{\partial\Delta^n}$ is <homotopic> to the <identity map>. It therefore induces the identity on reduced top <homology>. Yet it extends across the contractible <simplex>, whose reduced <homology> is zero, a contradiction.

Conversely, any such face-preserving boundary-valued <continuous map> would produce a counterexample by $F_i=\{x:g_i(x)=0\}$. These are closed, cover the <simplex>, contain their opposite <faces of a simplex> and have empty common <intersection>, because the $g_i$ sum to one. Thus the covering assertion and the absence of that map are equivalent.


No-retraction covering principle

Distance-function barycentric map

No-retraction covering principle

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No-retraction covering principle

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