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The projective $ \pi$-character bounds the order of a $ \pi$-base

Authors: István Juhász and Zoltán Szentmiklóssy
Journal: Proc. Amer. Math. Soc. 136 (2008), 2979-2984
MSC (2000): Primary 54A25, 54C10, 54D70
Published electronically: April 2, 2008
MathSciNet review: 2399066
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Abstract: 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 (Tychonov) 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 is a significant generalization of a celebrated result of Shapirovskii.

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Additional Information

István Juhász
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, POB 127, Budapest, H-1364 Hungary

Zoltán Szentmiklóssy
Affiliation: Department of Analysis, Eötvös Loránt University, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

Keywords: Projective $\pi $-character, order of a $\pi $-base, irreducible map
Received by editor(s): March 28, 2007
Received by editor(s) in revised form: June 17, 2007
Published electronically: April 2, 2008
Additional Notes: This research was supported by OTKA grant no. 61600.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.