Let $X$ be a topological space. Assume that the topology on $X$ is defined by the expansive sequence of compact subsets $$\ldots\subset K_{d-1}\subset K_d\subset K_{d+1}\subset \ldots\subset X, \: \: X=\bigcup_dK_d$$ i.e. we say that $C\subset X$ is closed if and only if $C\cap K_d$ is closed for any $d$. It seems quite reasonable that every $x\in X$ has a compact neighborhood, however I'm not able to prove it or to find a counterexample. Any help?
PS: Feel free of assuming $X$ Hausdorff.
Even for CW-complexes this is not true, which offers an easy way of constructing counterexamples. Any countable CW-complex is a direct limit of a sequence of finite subcomplexes, but only locally finite complexes are locally compact. For a minimal counterexample you can just take a single 0-cell and attach $\aleph_0$ 1-cells.