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The topology of almost everywhere continuous, approximately continuous functions

The topology of almost everywhere continuous, approximately continuous functions Acta Mathematica Academiae Seientiarum Hungaricae Tomus 37 (4), (1981), 317--328. THE TOPOLOGY OF ALMOST EVERYWHERE CONTINUOUS, APPROXIMATELY CONTINUOUS FUNCTIONS By T. NISHIURA (Detroit) O. Introduction At the outset we make the following agreements: 1. All functions are defined on R N. 2. If P is a statement about points of R ~ then E(P) will denote the subset of R n for which the statement P is true. 3. The Lebesgue measure is denoted by /~. Let ~ be a family of functions. As usual, 9~1(~r denotes the family of all functions which are pointwise limits of sequences of functions in s We shall concern ourselves with the family cg of continuous functions, the family d of approximately continuous functions [4] and the family 9~ of almost everywhere continuous functions. In [6], GRANDE considers, for R 1, the relationships between ~l(cg), N, (dBg~) and ~I(d)NNI(N). He derives a necessary condition which distinguishes Ni(dNN) and ~l(d)NN~(~) and further shows that N~(cd)# /~l(sCNr In the present paper we show sr ~ is associated with a completely regular, Hausdorff topology just as cg and ~' are. We show the condition of GRANDE [6] which distinguishes N~(~CN~) and NI(~r is a natural consequence http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

The topology of almost everywhere continuous, approximately continuous functions

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895131
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Abstract

Acta Mathematica Academiae Seientiarum Hungaricae Tomus 37 (4), (1981), 317--328. THE TOPOLOGY OF ALMOST EVERYWHERE CONTINUOUS, APPROXIMATELY CONTINUOUS FUNCTIONS By T. NISHIURA (Detroit) O. Introduction At the outset we make the following agreements: 1. All functions are defined on R N. 2. If P is a statement about points of R ~ then E(P) will denote the subset of R n for which the statement P is true. 3. The Lebesgue measure is denoted by /~. Let ~ be a family of functions. As usual, 9~1(~r denotes the family of all functions which are pointwise limits of sequences of functions in s We shall concern ourselves with the family cg of continuous functions, the family d of approximately continuous functions [4] and the family 9~ of almost everywhere continuous functions. In [6], GRANDE considers, for R 1, the relationships between ~l(cg), N, (dBg~) and ~I(d)NNI(N). He derives a necessary condition which distinguishes Ni(dNN) and ~l(d)NN~(~) and further shows that N~(cd)# /~l(sCNr In the present paper we show sr ~ is associated with a completely regular, Hausdorff topology just as cg and ~' are. We show the condition of GRANDE [6] which distinguishes N~(~CN~) and NI(~r is a natural consequence

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Jun 18, 2005

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