Let $(X, \mathcal{U})$ be a strongly Lindeloff uniform space ( it is complete separable) and $f:X\to X$ be a homeomorphism. A point $x\in X$ is called a fixed point of $f$ if $f(x)=x$. Denote by $Fix(f)$ the set of fixed points of $f$. Take $D\in \mathcal{U}$ a symmetric entourage and $D[x]=\{y\in X: (x, y)\in D\}$.

Is there a sequence $x_n\in \operatorname{Fix}(f)$ such that \begin{equation} \operatorname{Fix}(f)=\bigcup_{n\in\mathbb{N}}D[x_n]\cap \operatorname{Fix}(f)? \end{equation} Please help me to know it.


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