According to Wikipedia, 'finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures.' I have been studying the book 'Counterexamples in Topology' (by L. Steen & J. Seebach), and I have not found any explicit nontrivial 'counterexamples' per se given by finite topological spaces in this book.
The only nontrivial example I have thus far found of a counterexample given by a finite topological space is given in the article 'A counterexample in finite fixed point theory' by John R. Isbell (under the pseudonym 'H. C. Enos'). Isbell exhibited a finite topological space which is a counterexample to the following question:
Question (Lee Mohler, 1970): Given a topological space $X$ which is the union of closed subspaces $Y$ and $Z$ such that $Y$, $Z$, and $Y \cap Z$ have the fixed point property, does it follow that $X$ has the fixed point property?
It thus seems natural to ask: what are some other examples of interesting nontrivial counterexamples given by finite topological spaces?
$\endgroup$ 51 Answer
$\begingroup$The Sierpiński space is the smallest example of a topological space which is neither trivial nor discrete. It is also the smallest Kolmogorov ($T_0$) space that is not Hausdorff.
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