Open Access
2000/2001 Oscillation and ω-Primitives
J. Ewert, S. P. Ponomarev
Real Anal. Exchange 26(2): 687-702 (2000/2001).


We extend the results of [2], [6] in the case of topological spaces. It is shown that given an upper semicontinuous (USC) function $f:X\to [0,\infty)$ where $X$ is a massive first countable $T_1$-space satisfying some "neighborhood conditions", there exists $F:X\to [0, \infty)$ whose oscillation equals $f$ everywhere on $X$ (Theorem 2.1). The analogous result holds for USC functions $f:X\to [0, \infty]$ if, in addition, $X$ is a normal space (Theorem 2.4). A special metrizability criterion is established (Theorem 1.1). This is to show, by exhibiting corresponding examples, that the neighborhood conditions and massiveness do not imply that $X$ is Baire or metrizable. Among other related topics, sequences of $\omega$-primitives are discussed.


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J. Ewert. S. P. Ponomarev. "Oscillation and ω-Primitives." Real Anal. Exchange 26 (2) 687 - 702, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 27 June 2008

zbMATH: 1025.26002
MathSciNet: MR1844386

Primary: 26A15 , ‎54C30 , 54C99

Keywords: $\omega$-primitive , $\sigma$-discrete set , first countable space , massive space , ‎oscillation‎ , quasi-uniform convergence , Teichm\"uller-Tukey's lemma , upper semicontinuous function

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 2 • 2000/2001
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