Lower separation axioms via Borel and Baire algebras

Abstract

Let $\kappa$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{\kappa-Borel}$-space (resp. a $T_{\kappa-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $\kappa$-additive algebra of subsets of $X$ that contains all open sets (and all nowhere dense sets) in $X$. Each $T_1$-space is a $T_{\kappa-Borel}$-space and each $T_{\kappa-Borel}$-space is a $T_0$-space. On the other hand, $T_{\kappa-BP}$-spaces need not be $T_0$-spaces. We prove that a topological space $X$ is a $T_{\kappa-Borel}$-space (resp. a $T_{\kappa-BP}$-space) if and only if for each point $x\in X$ the singleton $\{x\}$ is the intersection of a closed set and a $G_{<\kappa}$-set in $X$ (resp. $\{x\}$ is either nowhere dense or a $G_{<\kappa}$-set in $X$). Also we present simple examples distinguishing the separation axioms $T_{\kappa-Borel}$ and $T_{\kappa-BP}$ for various infinite cardinals $\kappa$, and we relate the axioms to several known notions, which results in a quite regular two-dimensional diagram of lower separation axioms.