When is a point Borel?

Peter Wamer Harley, III and George Frank McNulty

Vol. 80 (1979), No. 1, 151–157

DOI: 10.2140/pjm.1979.80.151

Abstract

Let X be a topological space. We investigate the question: When is a point (of X) Borel? In relation to this, we establish the equivalence of (a) Each point (singleton) is Borel, (b) Each point is the intersection of closed set and a G_δ, (c) The derived set of each point is Borel, (d) The derived set of each point is an F_σ, (e) The derived set of each subset is Borel, and (f) The derived set of each subset is an F_σ. Conditions (a), (b), (c), and (d) are also equivalent for a fixed point. As a separation axiom (a) is shown to lie strictly between T₁ and T₀. A number of examples are given and the work of other authors discussed.

Mathematical Subject Classification 2000

Primary: 54D10

Secondary: 54H05

Milestones

Received: 29 July 1977

Published: 1 January 1979