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A space X is almost star countable (weakly star countable) if for each open cover U of X there exists a countable subset F of X such that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\bigcup {_{x \in F}\overline {St\left( {x,U} \right)} } = X$ \end{document} (respectively, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\overline {\bigcup {_{x \in F}} St\left( {x,U} \right)} = X$ \end{document} . In this paper, we investigate the relationships among star countable spaces, almost star countable spaces and weakly star countable spaces, and also study topological properties of almost star countable spaces.

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