View source for Pseudo-basis

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''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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====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|Regular space]]) if and only if it has a closed pseudo-basis (in the second sense).