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

##### 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
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##### 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
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