Projective $\pi$-character bounds the order of a $\pi$-base

István Juhász, and Zoltán Szentmiklóssy

All spaces below are Tychonov. We define the projective $\pi$-character $p\,\pi\chi (X)$ of a space $X$ as the supremum of the values $\pi\chi(Y)$ where $Y$ ranges over all continuous images of $X$. Our main result says that every space $X$ has a $\pi$-base whose order is $\le p\,\pi\chi (X)$, that is every point in $X$ is contained in at most $p\,\pi\chi (X)$-many members of the $\pi$-base. Since $p\,\pi\chi (X) \le t(X)$ for compact $X$, this provides a significant generalization of a celebrated result of Shapirovskii.

Key words and phrases: Projective pi-character, order of a pi-base, irreducible map

2000 Mathematics Subject Classification: 54A25, 54C10, 54D70

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