A note on Volterra and Baire spaces

Document Type: Research Paper

Authors

College of Applied Science‎, ‎Beijing University of Technology‎, ‎Beijing 100124‎, ‎China

Abstract

 In Proposition 2.6 in (G‎. ‎Gruenhage‎, ‎A‎. ‎Lutzer‎, ‎Baire and Volterra spaces‎, ‎\textit{Proc‎. ‎Amer‎. ‎Math‎. ‎Soc.} {128} (2000)‎, ‎no‎. ‎10‎, ‎3115--3124) a condition that‎ ‎every point of $D$ is $G_\delta$ in $X$ was overlooked‎. ‎So we‎ ‎proved some conditions by which a Baire space is equivalent to a‎ ‎Volterra space‎. ‎In this note we show that if $X$ is a‎ ‎monotonically normal $T_1$-space with countable pseudocharacter ‎and $X$ has a $\sigma$-discrete dense subspace $D$‎, ‎then $X$ is a‎ ‎Baire space if and only if $X$ is Volterra‎. ‎We show that if $X$‎ ‎is a metacompact normal sequential $T_1$-space and $X$ has a‎ ‎$\sigma$-closed discrete dense subset‎, ‎then $X$ is a Baire space‎ ‎if and only if $X$ is Volterra‎. ‎If $X$ is a generalized ordered‎ ‎(GO) space and has a $\sigma$-closed discrete dense subset‎, ‎then‎ ‎$X$ is a Baire space if and only if $X$ is Volterra‎. ‎And also some‎ ‎known results are generalized‎.

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Volume 41, Issue 6
November and December 2015
Pages 1445-1452
  • Receive Date: 22 May 2014
  • Revise Date: 10 September 2014
  • Accept Date: 10 September 2014