On the \(\omega \)-primitive. (English) Zbl 0987.54031

Duszynski, Zbigniew; Grande, Zbigniew; Ponomarev, Stanislaw P.

Math. Slovaca 51, No. 4, 469-476 (2001).

A new class of metric spaces is introduced, so-called massive spaces. The following definitions are assumed: a metric space \(X\) is said to be \(\sigma \)-discrete at a point \(x \in X\) if there is a \(\sigma \)-discrete neighbourhood of \(x\); and \(X\) is called massive if it is not \(\sigma \)-discrete at each of its points. The main result is given in Theorem 2, namely: If \(X\) is a massive metric space and \(f: X \to [0,\infty ]\) is upper semicontinuous, then there exists a function \(F: X \to [0, \infty)\) at most in the Baire class two such that the oscillation of \(F\) equals \(f\).
Since each dense-in-itself metric Baire space is massive, this theorem extends the result of P. Kostyrko [ibid. 30, 157-162 (1980; Zbl 0441.54004)].

Reviewer: Janina Ewert (Słupsk)

MSC:

54E35	Metric spaces, metrizability
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C30	Real-valued functions in general topology

Keywords:

metric space; \(\sigma \)-discrete set oscillation; Teichmüller-Tukey’s lemma; Baire class

Citations:

Zbl 0441.54004

References:

[1]	ENGELKING R.: General Topology. Vol. 1,2, Warsaw, PWN, 1989. · Zbl 0684.54001
[2]	EWERT J.-PONOMAREV S. P.: Oscillation and to-primitives. Real Anal. Exchange 26 (2000/2001) · Zbl 1025.26002
[3]	KELLEY J. L.: General Topology. D. van Nostrand Co., Inc., New York, 1955. · Zbl 0066.16604
[4]	KOSTYRKO P.: Some properties of oscillation. Math. Slovaca 30 (1980), 157-162. · Zbl 0441.54004