Is it true that in a locally compact (not necessarily Hausdorff) space, every point has a neighbourhood bases consisting of precompact subsets? (precompact means closure is compact).
In addition, if the space is also Hausdorff, then we have a local base consisting of compact neighbourhoods.
The first assertion is easy to see in a Hausdorff space as compact subsets of a Hausdorff space is closed. However, I cannot see this without the Hausdorff condition.
Locally compact means every point has a neighbourhood (a set containing an open set containing that point) which is compact.