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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.28092 |
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| _version_ | 1866918476268961792 |
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| author | Carlson, Nathan |
| author_facet | Carlson, Nathan |
| contents | In 1967 Hajnal and Juh{á}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 $π$-base".
Given a point $p$ in a topological space $X$, a \emph{local} $π$-\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 $π$-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 $π$-base \emph{centered}. A centered local $π$-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 $π$-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 $π$-base. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_28092 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On centered local $π$-bases Carlson, Nathan General Topology 54D10, 54A25 In 1967 Hajnal and Juh{á}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 $π$-base". Given a point $p$ in a topological space $X$, a \emph{local} $π$-\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 $π$-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 $π$-base \emph{centered}. A centered local $π$-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 $π$-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 $π$-base. |
| title | On centered local $π$-bases |
| topic | General Topology 54D10, 54A25 |
| url | https://arxiv.org/abs/2604.28092 |