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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