In this question we consider only $T_2$ spaces. A space is pseudocompact if evey continuous $f:X\to \mathbb{R}$ is bounded; it is feebly compact if every locally finite open cover $\{A_i\}\not\ni\emptyset$ is finite. It is known that, if $X$ is $T_{3+\frac12}$, pseudocompact is equivalent to feebly compact (see "General Topology", Engelking, thm 3.10.22). It is also easy to see that feebly compact implies pseudocompact withouth this hypotesis: given an unbounded continuous function $f$, $\{f^{-1}((n-1,n+1))\}_{n\in\mathbb{Z}}-\{\emptyset\}$ shows that $X$ is not feebly compact.
I have read somewhere (can't recall where) that in general pseudocompact does not imply feebly compact. I tried constructing a counterexample considering a countable connected $T_2$ space (which is clearly pseudocompact, being strongly connected), but I don't see how to construct a locally finite infinite open cover. So my question is: what is an example of a pseudocompact not feebly compact space? Does my idea work?
Looking in the standard reference on such matters: Porter and Woods, Extensions and Absolutes of Hausdorff Spaces, we find such an example in exercise 1U at p. 66:
Sketch: let $X(n), n \in \Bbb N$ be a countably infinite partition of $[0,1]$ into dense subsets, let $\tau$ be the usual topology on $[0,1]$ and let $\sigma$ be the topology on $[0,1]$ for which the family $$\tau \cup \{X(2n-1) \mid n \in \Bbb N\} \cup \{X(2n-1) \cup X(2n) \cup X(2n+1)\mid n \in \Bbb N\}$$ is a subbase.
Let $Y=([0,1], \sigma)$, then it is claimed that $Y$ is Hausdorff and not regular, pseudocompact but not feebly compact.
See the book for more full hints, no solution or external reference (paper) is given. But the source seems to be example 3.5 in this paper, which adds explanations and many equivalent formulations as well.