A Particular Point Topology is not compact, is path-connected but what about Fixed Point Property? Does it have fixed point property? If so how?
I have been told that it should have Fixed Point Property. But I do not know how to Prove it.
2026-04-03 12:58:12.1775221092
Does Particular Point Topology has Fixed Point Property?
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Hint: Suppose that $X$ is given the particular point topology with particular point $p$, and let $f : X \to X$ be continuous. If $f(p) = p$, we're done. If $f(p) \neq p$, then $\{ f(p) \}$ is closed in $X$, and so $f^{-1} [ \{ f(p) \} ]$ is also closed in $X$. What are the closed subsets of $X$ like?