On centered local $\pi$-bases
In 1967 Hajnal and Juh{\'a}sz showed that the cardinality of a first-countable Hausdorff space with the countable chain condition has cardinality at most $\mathfrak{c}$, the cardinality of the real line. We give an improvement of this celebrated theorem by replacing ``first-countable" with the weaker condition ``each point has a countable centered local $\pi$-base". Given a point $p$ in a topological space $X$, a \emph{local} $\pi$-\emph{base} $\scr{B}$ at $p$ acts like a neighborhood base at $p$ except that $p$ may not be in any member of $\scr{B}$. A local $\pi$-base $\scr{B}$ has the \emph{finite intersection property} if any finite intersection of members of $\scr{B}$ is nonempty. We call this type of local $\pi$-base \emph{centered}. A centered local $\pi$-base behaves even more like a neighborhood base in a sense. A space has the \emph{countable chain condition} if every family of pairwise disjoint open sets is countable. We also improve a theorem of Pospi{\v s}il from 1937 using centered local $\pi$-bases. As is customary, examples are given to demonstrate these improvements are strict. Compact Hausdorff spaces are also explored in this connection, along with variations on the notion of a centered local $\pi$-base.