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| In ], a ] ''X'' has '''the fixed point property''' if all ]s from ''X'' to ''X'' have a ]. | In ], a ] ''X'' has '''the fixed point property''' if all ] ]s from ''X'' to ''X'' have a ]. | ||
| ==Examples== | |||
| ===The closed interval=== | |||
| The ] has the fixed point property: Let ''f: → '' be a mapping. If ''f(0) = 0'' or ''f(1) = 1'', then our mapping has a fixed point at 0 or 1. If not, then ''f(0) > 0'' and ''f(1) - 1 < 0''. Thus the function ''g(x) = f(x) - x'' is a continuous real valued function which is positive at ''x=0'' and negative at ''x=1''. By the ], there is some point ''x<sub>0</sub>'' with ''g(x<sub>0</sub>) = 0'', which is to say that ''f(x<sub>0</sub>) - x<sub>0</sub> = 0'', and so ''x<sub>0</sub>'' is a fixed point. | |||
| The ] does ''not'' have the fixed point property. The mapping ''f(x) = x<sup>2</sup>'' has no fixed point on the interval (0,1). | |||
| ===The closed disc=== | |||
| The closed interval is a special case of the ], which in any dimension has the fixed point property by the ]. | |||
| {{topology-stub}} | {{topology-stub}} | ||
Revision as of 13:08, 11 January 2007
In mathematics, a topological space X has the fixed point property if all continuous mappings from X to X have a fixed point.
Examples
The closed interval
The closed interval has the fixed point property: Let f: → be a mapping. If f(0) = 0 or f(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then f(0) > 0 and f(1) - 1 < 0. Thus the function g(x) = f(x) - x is a continuous real valued function which is positive at x=0 and negative at x=1. By the intermediate value theorem, there is some point x0 with g(x0) = 0, which is to say that f(x0) - x0 = 0, and so x0 is a fixed point.
The open interval does not have the fixed point property. The mapping f(x) = x has no fixed point on the interval (0,1).
The closed disc
The closed interval is a special case of the closed disc, which in any dimension has the fixed point property by the Brouwer fixed point theorem.
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