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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