I wonder if there is a compact and locally Hausdorff space $X$ which is not locally compact, in the sense that every point has a neighborhood base consisting of compact sets.
A space is called locally Hausdorff if every point has a neighborhood which is a Hausdorff space under the subspace topology. Such a space need not be Hausdorff. However, if each point has a closed Hausdorff neighborhood, then the space is Hausdorff. An example is the real line with two origins. If one restricts this space between $-1$ and $1$, it becomes compact. Unfortunately, it is also locally compact.
If $X$ is Hausdorff and compact, then it is also locally compact, so the Hausdorff neighborhoods must all be proper subsets of $X$.
Does anyone have an idea?