the topologist’s sine curve has the fixed point property

The typical example of a connected space that is not path connected (the topologist’s sine curve) has the fixed point property.

Let X1={0}×[-1,1] and X2={(x,sin(1/x)):0<x1}, and X=X1X2.

If f:XX is a continuous mapMathworldPlanetmath, then since X1 and X2 are both path connected, the image of each one of them must be entirely contained in another of them.

If f(X1)X1, then f has a fixed point because the intervalMathworldPlanetmathPlanetmath has the fixed point property. If f(X2)X1, then f(X)=f(cl(X2))cl(f(X2))X1, and in particular f(X1)X1and again f has a fixed point.

So the only case that remains is that f(X)X2. And since X is compactPlanetmathPlanetmath, its projection to the first coordinate is also compact so that it must be an interval [a,b] with a>0. Thus f(X) is contained in S={(x,sin(1/x)):x[a,b]}. But S is homeomorphic to a closed interval, so that it has the fixed point property, and the restriction of f to S is a continuous map SS, so that it has a fixed point.

This proof is due to Koro.

Title the topologist’s sine curve has the fixed point property
Canonical name TheTopologistsSineCurveHasTheFixedPointProperty
Date of creation 2013-03-22 16:59:37
Last modified on 2013-03-22 16:59:37
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 6
Author Mathprof (13753)
Entry type Proof
Classification msc 55M20
Classification msc 54H25
Classification msc 47H10