In A first countable hemicompact space is locally compact we can see a proof that every first-countable hemicompact space is weakly locally compact.
If first-countability is dropped, is this still true? In other words, is there a hemicompact space that is not weakly locally compact? pi-base currently does not have an example of this.
Let $X=\{x_n:n<\omega\}$ be a countable and anticompact (every compact subset is finite) space. Then $X$ is hemicompact (T000291): let $K_n=\{x_i:i<n\}$, then every compact is finite and thus contained in some $K_n$.
Then if such a space contains a point with no finite neighborhoods, such as the Arens-Fort Space or Appert space, the space cannot be weakly locally compact.
Going over the linked page 20 in "Counterexamples in Topology" by Lynn A. Steen and J. Arthur Seebach, I assume "weakly locally compact" is just "locally compact". There is an example of a hemicompact, but not locally compact space found in Chapter 4, Section 2, Example 10 and Exercise 8 in "Introduction to General Topology" (See here.) by K. D. Joshi.
