I've been reviewing some local definitions of topological spaces, like locally compact space, locally connected space, etc. It seems that such properties can be phrased in terms of a local base, but I was wondering whether these properties can be phrased in terms of a topological basis.
For example, can we say that $X$ is locally compact if and only if it has a compact basis (consisting of compact sets)?
You can say that a Hausdorff space $X$ is locally compact iff it has a base of relatively compact sets (so sets whose closure is compact), not of compact sets, because sets in a base have to open by definition (and in many spaces this implies they cannot be compact too, e.g. in Hausdorff connected spaces).
For locally connected spaces we can say that it has a base of connected subsets (it sort of depends on your definition of local connectedness, but this is true in most cases).
This is because the union of all local bases is a base and every base contains a local base for each of its points.