TITLE: Is every locally compact Hausdorff space paracompact?
QUESTION [11 upvotes]: It seems likely that for any open cover, we can construct a locally finite refinement using the local compactness of the space. I can't figure out how to work the construction though, and I'm not yet convinced that there is no counterexample.

REPLY [3 votes]: $\pi$-Base is an online encyclopedia in the spirit of Steen and Seebach's Counterexamples in Topology. According to its database, the following spaces are locally compact Hausdorff but not paracompact. (You can view the search result to learn more.)
$[0,\Omega) \times I^I$
Deleted Tychonoff Plank
Open Ordinal Space $[0,\Omega)$
Rational Sequence Topology
The Long Line
The Open Long Line
Thomas' Plank