Is every locally compact Hausdorff space paracompact?
Solution
No. A counterexample is \[ S_\Omega=[0,\Omega), \] the space of countable ordinals with the order topology. It is Hausdorff and locally compact: for every \(\alpha<\Omega\), a sufficiently small closed order interval about \(\alpha\) is a compact ordinal interval. But \(S_\Omega\) is not paracompact by E-94C75(b). Hence local compactness plus Hausdorffness does not imply paracompactness.