On a generalization of the density topology on the real line. (English) Zbl 1151.54005

Wojdowski, Wojciech

Real Anal. Exch. 33(2007-2008), No. 1, 199-214 (2008).

The author studies his generalization of the density topology defined as follows, see [Real Anal. Exch. 32(2006–2007), No. 2, 349–358 (2007; Zbl 1135.28001)]: call \(x\) an \(\mathcal{A}_d\)-density point of a subset \(A\) of \(\mathbb{R}\) if for every sequence \(\langle t_n\rangle_n\) of reals that diverges to \(\infty\) there are a set \(B\subseteq[-1,1]\) with \(0\) as a density point and a subsequence \(\langle t_{n_m}\rangle_m\) such that the characteristic functions of \((t_{n_m}\cdot(A-x))\cap [-1,1]\) converge almost everywhere to that of \(B\). The \(\mathcal{A}_d\)-density topology consists of those sets every point of which is an \(\mathcal{A}_d\)-density point. Among the results proved are:
a function is measurable iff it is \(\tau_{\mathcal{A}_d}\)-continuous at almost every point; the topology \(\tau_{\mathcal{A}_d}\) is completely regular but not normal and every \(\tau_{\mathcal{A}_d}\)-continuous function is of Baire class \(1\).

Reviewer: K. P. Hart (Delft)

MSC:

54A10	Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
28A05	Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
26A21	Classification of real functions; Baire classification of sets and functions

Keywords:

density point; density topology

Citations:

Zbl 1135.28001