Each topological space $A$ with fixedpoint property is connected (all clopen subsets are trivial). This is an analog of Rice theorem (all decidable subsets are trivial). Suppose, we have a space $A$ with the following *property: each closed subspace of $A$ has fixedpoint property. For example, $A$ may be a Scott domen. Then every two closed subsets of $A$ have a nonempty intersection! If $A$ is compact, all closed subsets of $A$ have a nonempty intersection. Did anyone investigate *property? For example, closed subsets of $A$ may have fixedpoint property because they are "generalized retracts" of $A$ in some sence.
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$\begingroup$ Moderator added tag "gn.general topology", but I am not sure this is a general topological question. $\endgroup$ – George Cherevichenko Mar 31 '18 at 22:21