In Wilder's Topology of Manifolds, the following is stated on p. 43:
"It is well known that not every countably compact, separable, connected space is compact."
Hmm . . . I'm not sure just how well-known this is. He also assumes the spaces in question are $T_1$. Since a google search only pulled up a considerably more advanced paper, I figured MSE might be a good place to have this example. Does anyone know a relatively easy one?
Really appreciate it! I will scour Seebach a bit later if nobody wants to grab this problem.
An example mentioned in this survey paper by Nyikos is $X = \omega_1$ with the topology generated by the cofinite sets and the initial segments $[0,\alpha)$, $\alpha < \omega_1$. This is not Hausdorff, but does obey the other properties of $T_1$, connectedness, countable compactness and non-compactness, as is easy to see. So this gives a ZFC example of the non-implication.
The survey paper gives a nice overview of how hard it is to construct Hausdorff first-countable such spaces. Examples are known (like Ostaszewski spaces) but need an additional set-theoretical assumption. Connected such spaces are not yet known (most such set theory based examples tend to be scattered or zero-dimensional, quite the opposite of connected).