Let $(X,d)$ a bounded metric space. We say that $X$ has the complete invariance property (shortly, the CIP) if for each non-empty and closed $A\subset X$ there is a continuous $f:X\rightarrow X$ such that $A$ is the set of fixed points of $X$.
Assume that there is a sequence of compact subsets of $X$, put $K_{n}$, such that $d_{H}(K_{n},X)\leq 1/n$ (here, $d_{H}$ is the Hausdorff metric) and $K_{n}$ has the CIP. Under these conditions, Has $X$ the CIP?
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