compact neighbourhoods.
:3. every point of X has a local base of compact neighbourhoods.
:4. every point of X has a local base of closed compact neighbourhoods.
:5. X is Hausdorff and satisfies any (or equivalently, all) of the previous conditions.

Logical relations among the conditions:
* Each condition implies (1).
* Conditions (2), (2´), (2´´) are equivalent.
* Neither of conditions (2), (3) implies the other.
* Condition (4) implies (2) and (3).
* Compactness implies conditions (1) and (2),