Proposition 7

If $A$ is a relatively quasi-compact subset of a topological space $X$, then every filter base on $A$ has a cluster point in $X$.

For if $A \subset K$, where $K$ is a quasi-compact subset of $X$, then every filter base on $A$ has a cluster point in $K$.

The converse of this proposition is not valid without restriction on $X$ (Exercise 22).