Fixed point sets of 𝐿𝐶^{∞}, 𝐶^{∞} continua

Martin, John R.

doi:10.1090/S0002-9939-1981-0593482-4

Fixed point sets of $\textrm {LC}^{\infty }$, $C^{\infty }$ continua
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Proc. Amer. Math. Soc. 81 (1981), 325-328 Request permission

Abstract:

A space $X$ is said to have the complete invariance property (CIP) if every nonempty closed subset of $X$ is the fixed point set of some self-map of $X$. An example is given to show that there is an $L{C^\infty }$, ${C^\infty }$ continuum $X$ which does not have CIP. Moreover, $X$ is a wedge of two $L{C^\infty }$, ${C^\infty }$ continua each of which has CIP.

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