Vol. 80, No. 1, 1979

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When is a point Borel?

Peter Wamer Harley, III and George Frank McNulty

Vol. 80 (1979), No. 1, 151–157

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 Gδ, (c) The derived set of each point is Borel, (d) The derived set of each point is an Fσ, (e) The derived set of each subset is Borel, and (f) The derived set of each subset is an Fσ. 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
Received: 29 July 1977
Published: 1 January 1979
Peter Wamer Harley, III
George Frank McNulty