Lower separation axioms via Borel and Baire algebras

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F18%3A10390725" target="_blank" >RIV/00216208:11320/18:10390725 - isvavai.cz</a>
Result on the web
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DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Lower separation axioms via Borel and Baire algebras
Original language description
Let κ be an infinite regular cardinal. We define a topological space X to be a T _{κ-Borel}-space (resp. a T_{κ-BP}-space) if for every x ELEMENT OF X the singleton {x} belongs to the smallest κ-additive algebra of subsets of X that contains all open sets (and all nowhere dense sets) in X. Each T_1-space is a T_{κ-Borel}-space and each T_{κ-Borel}-space is a T_0-space. On the other hand, T_{κ-BP}-spaces need not be T_0-spaces. We prove that a topological space X is a T_{κ-Borel}-space (resp. a T_{κ-BP}-space) if and only if for each point x ELEMENT OF X the singleton {x} is the intersection of a closed set and a G_{<κ}-set in X (resp. {x} is either nowhere dense or a G_{<κ}-set in X). Also we present simple examples distinguishing the separation axioms T_{κ-Borel} and T_{κ-BP} for various infinite cardinals κ, and we relate the axioms to several known notions, which results in a quite regular two-dimensional diagram of lower separation axioms.
Czech name
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Czech description
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Project
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Continuities
S - Specificky vyzkum na vysokych skolach<br>I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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