Fixed-point theorem: Difference between revisions

Revision as of 11:06, 18 October 2022

Revision as of 11:07, 18 October 2022

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The above technique of iterating a function to find a fixed point can also be used in [[set theory]]; the [[fixed-point lemma for normal functions]] states that any continuous strictly increasing function from [[ordinal number|ordinals]] to ordinals has one (and indeed many) fixed points.

Every [[closure operator]] on a [[poset]]posset has many fixed points; 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.

Every [[involution (mathematics)|involution]] on a [[finite set]] with an odd number of elements has a fixed point; more generally, for every involution on a finite set of elements, the number of elements and the number of fixed points have the same [[parity (mathematics)|parity]]. [[Don Zagier]] used these observations to give a one-sentence proof of [[Fermat's theorem on sums of two squares]], by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares. Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form.<ref>{{citation