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$ \mathcal{L}$-realcompactifications as epireflections


Authors: H. L. Bentley and S. A. Naimpally
Journal: Proc. Amer. Math. Soc. 44 (1974), 196-202
MSC: Primary 54D60
DOI: https://doi.org/10.1090/S0002-9939-1974-0365489-X
MathSciNet review: 0365489
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Abstract: If $ \mathcal{L}$ is a countably productive normal base on a Tychonoff space $ X$, then $ \eta (X,\mathcal{L})$ is an $ {\mathcal{L}_ \ast }$-realcompact extension of $ X$. R. A. Alo and H. L. Shapiro thus generalized the Hewitt realcompactification of $ X$. In the following paper, we extend this construction to $ {T_1}$-spaces and show that it is an epireflection functor on an appropriate category. We are thus concerned with the question of the extendibility of a continuous map $ f:X \to Y$ to a continuous map $ g:\eta (X,{\mathcal{L}_X}) \to \eta (Y,{\mathcal{L}_Y})$. We derive necessary and sufficient conditions therefor in the case when $ {\mathcal{L}_Y}$ is a nest generated intersection ring on $ Y$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0365489-X
Keywords: Separating base, normal base, nest generated intersection ring, $ \mathcal{L}$-realcompactification, proximity, cluster, $ \omega $-contiguity, epireflection, $ \omega $-map, extension of maps
Article copyright: © Copyright 1974 American Mathematical Society

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