Pseudo-basis

of a topological space $X$

A family of sets open in $X$ and such that each point of $X$ is the intersection of all elements in the family containing it. A pseudo-basis exists only in spaces all singletons of which are closed (i.e. in $T_1$-spaces). If a $T_1$-space with basis $\mathcal B$ is endowed with a stronger topology, then $\mathcal B$ is no longer a basis of the new topological space but remains a pseudo-basis of it. In particular, a discrete space of the cardinality of the continuum, which does not have a countable basis, has a countable pseudo-basis. However, for Hausdorff compacta (i.e. compact Hausdorff spaces) the presence of a countable pseudo-basis implies the existence of a countable basis.

References

[1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Comments

The term pseudo-basis is also used in two other ways, as follows.

A collection of non-empty open sets (in a topological space $X$) such that every non-empty open set of $X$ contains one of these is also sometimes called a pseudo-basis, although the term $\pi$-basis is favoured nowadays.

Another use of "pseudo-basis" is for a collection $\mathcal A$ of subsets of a topological space $X$ such that for every open set $O$ and every point $x$ of $O$ there is an element $A$ of $\mathcal A$ such that

$$x\in\operatorname{int}A\subset A\subset O.$$

Hence a topological space is regular (cf. Regular space) if and only if it has a closed pseudo-basis (in the second sense).