This group gathers theorems that guarantee a continuous map has a point left unmoved, one of the most widely applied families of results in topology and analysis. It covers Brouwer's fixed-point theorem for continuous maps of a ball to itself, the Leray-Schauder and Schauder extensions of that idea to infinite-dimensional spaces, and the related fixed-point results of Lefschetz and Kakutani that carry the same guarantee into algebraic topology and game theory.
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Fixed-Point Theorems
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Brouwer Fixed-Point Theorem (Wikipedia)
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Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L.
Schauder fixed point theorem, Wikipedia
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The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension.
Lefschetz fixed-point theorem (Wikipedia)
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In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact triangulable topological space X {\displaystyle X} to itself by means of traces of the induced mappings on the homology groups of X {\displaystyle X} .
Kakutani Fixed-Point Theorem (Wikipedia)
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In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions.
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