Each topological space $A$ with fixed-point 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 fixed-point 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 fixed-point property because they are "generalized retracts" of $A$ in some sence.