Abstract

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

Highlights

  • Ðàçîì ç óçàãàëüíåííÿìè òåîðåìè Ñåðïiíñüêîãî íà âèïàäîê àáñòðàêòíèõ ïðîñòîðiâX , Y i Z ïðèðîäíî âèíèêà òàêîæ ïèòàííÿ ïðî ìîæëèâiñòü ïîñëàáëåííÿ óìîâ íàðiçíîíåïåðåðâíîñòi

  • Îäåðæàíî íîâi óçàãàëüíåííÿ òåîðåìè Ñåðïiíñüêîãî ïðî îäíîçíà÷íó âèçíà÷åíiñòü íàðiçíî íåïåðåðâíîôóíêöiíà äîáóòêó ñâîìè çíà÷åííÿìè íà äîâiëüíié âñþäè ùiëüíié ìíîæèíi

  • separately continuous function dened on the plane R2 is determined

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Summary

Ðàçîì ç óçàãàëüíåííÿìè òåîðåìè Ñåðïiíñüêîãî íà âèïàäîê àáñòðàêòíèõ ïðîñòîðiâ

X , Y i Z ïðèðîäíî âèíèêà òàêîæ ïèòàííÿ ïðî ìîæëèâiñòü ïîñëàáëåííÿ óìîâ íàðiçíîíåïåðåðâíîñòi. À ñàìå, ìîâà éäå ïðî îäåðæàííÿ òåîðåì ïðî îáîâ'ÿçêîâó ðiâíiñòü âiäîáðàæåíü f : X × Y → Z i g : X × Y → Z ç ïåâíîãî êëàñó, ÿêùî âîíè çáiãàþòüñÿ íà äåÿêié ùiëüíié â X × Y ìíîæèíi. Òàêîãî ñîðòó äîñëiäæåííÿ (äëÿ âiäîáðàæåíü áàãàòüîõ çìiííèõ) áóëî ïðîâåäåíî ó ðîáîòi [6], äå áóëî îäåðæàíî íàñòóïíèé ðåçóëüòàò. Íåõàé X áåðiâñüêèé ïðîñòið, Y òîïîëîãi÷íèé ïðîñòið çi çëi÷åííèì π -õàðàêòåðîì, Z óðèñîíîâèé ïðîñòið, A ⊆ X × Y ùiëüíà â X × Y ìíîæèíà, f : X ×. Çàçíà÷èìî, ùî, íåçâàæàþ÷è íà äîñèòü øèðîêå ðîçìàòòÿ óìîâ i òåðìiíîëîãi÷íi âiäìiííîñòi, ìåòîäè äîâåäåííÿ òåîðåì 2, 3 i 4, â öiëîìó, 1 ïîäiáíèìè. Ìè ïðîàíàëiçó1ìî, äëÿ ÿêèõ êëàñiâ ïðîñòîðiâ âèêîíó1òüñÿ òåîðåìà Ñåðïiíñüêîãî äëÿ íàðiçíî íåïåðåðâíèõ ôóíêöié i íàâåäåìî äåÿêi ïðèêëàäè

Îäíîñòàéíi âëàñòèâîñòi âiäîáðàæåíü äâîõ âiäîáðàæåíü
Ãîðèçîíòàëüíà ñëàáêà êâàçiíåïåðåðâíiñòü âiäíîñíî áàçè
Óçàãàëüíåííÿ òåîðåìè Ñåðïiíñüêîãî
Òðiéêè Ñåðïiíñüêîãî i âèïàäîê íàðiçíî íåïåðåðâíèõ âiäîáðàæåíü
Âiäêðèòi ïèòàííÿ
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