Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$.
The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the only continuous maps $f:X\to X$ with fixed points are the identity map $\text{id}_X:X\to X$, and the constant maps.
The only continuous self-maps of strongly rigid spaces are the identity and the constant maps, so they trivially have the AFPP.
Q. Assume that $(X,\tau)$ is Hausdorff and it has the AFPP. Does this imply that $(X,\tau)$ is strongly rigid?
Best Answer
Probably, the discrete $\{0,1\}$ is not the counterexample Dominic van der Zypen expected to see :)
A more elaborate CH-example of a AFPP but not strongly rigid space was constructed by van Mill:
Theorem 4.1 (van Mill, 1983). Under Continuum Hypothesis there exists a non-trivial metrizable separable connected locally connected Boolean topological group $X$ such that each continuous self-map $f:G\to G$ is either a translation or a constant map.
The space $G$ has AFFP but is not strongly rigid.
Remark. In the van Mill's proof the Continuum Hypothesis is used in combination with the following classical result of Sierpiński:
Theorem (Sierpiński, 1921). For any countable partition of the unit interval into closed subsets exactly one set of the partition is non-empty.
Motivated by this Sierpiński Theorem we can ask about the smallest infinite cardinality $\acute{\mathfrak n}$ of a partition of the unit interval into closed non-empty subsets. It is clear that $\acute{\mathfrak n}\le\mathfrak c$. The Sierpinski Theorem guarantees that $\omega_1\le\acute{\mathfrak n}$. This inequality can be improved to $\mathfrak d\le\acute{\mathfrak n}\le\mathfrak c$.
It seems that the proof of van Mill's Theorem actually yields more:
Theorem. Under $\acute{\mathfrak n}=\mathfrak c$ (which follows from $\mathfrak d=\mathfrak c$) there exists a non-trivial metrizable separable connected locally connected Boolean topological group $X$ such that each continuous self-map $f:G\to G$ is either a translation or a constant map.