In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological spaceX to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some imputed multiplicity at a fixed point. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).
In mathematics, the Lefschetzfixed-pointtheorem is a formula that counts the number of fixedpoints of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X.
Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space.
Lefschetz's focus was not on fixedpoints of mappings, but rather on what are now called coincidence points of mappings.
In mathematics, a fixed-pointtheorem is a result saying that a function F will have at least one fixedpoint (a point x for which F(x) = x), under some conditions on F that can be stated in general terms.
Every lambda expression has a fixedpoint, and a fixedpoint combinator is a "function" which takes as input a lambda expression and produces as output a fixedpoint of that expression.
Every closure operator on a poset has many fixedpoints; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.
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