Abstract
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.