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中文摘要：
借助关于子基的内部和闭包给出了β半开集和β半闭集的定义，进而定义了β半内部和β半闭包，研究了它们的基本性质和相互联系；并给出β半开集（β半闭集）的等价刻画；同时，还引入关于子基的半连续映射，并给出它的等价刻画。
英文摘要：
This paper first gives the definitions of β-semi-open sets and β-semi-close sets with the help ot the interior and the closure relative to a subbase and it defines the conceptions of β-semi-interior and β-semi-closure. Furthermore, it studies their basic properties and mutual relations, and gives their equivalent characterizations. Finally, it introduces the definition of semi-continuous maps relative to a subbase, and gives its equivalent characterizations.
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