Cosmic approximate limits and fixed points

Segal, J.; Watanabe, T.

doi:10.1090/S0002-9947-1992-1145962-2

Cosmic approximate limits and fixed points
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by J. Segal and T. Watanabe

Trans. Amer. Math. Soc. 333 (1992), 1-61

DOI: https://doi.org/10.1090/S0002-9947-1992-1145962-2

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Abstract:

We define a condition for approximate inverse systems which implies that the limit of the system has the fixed point property. Moreover, this condition is categorical in the approximate shape category. We investigate the class of complex projective $n$-space like continua with respect to the fixed point property by means of this condition. As a further application we show that the hyperspace $C(X)$ of nonempty subcontinua of an arc-like or circle-like Hausdorff continuum $X$ has the fixed point property. We also prove that ${2^X}$ and $C(X)$ have the fixed point property for any locally connected Hausdorff continuum $X$.

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