论文标题
在$α$ -COMPACT的拓扑空间上
On Countably $α$-Compact Topological Spaces
论文作者
论文摘要
在本文中，提出并证明了一些$α$ compact拓扑空间的某些功能。解释了$α$％compact，Tychonoff和$α$ -HAUSDORFF的空间之间的连接。如果有有限的空间，则该空间是$α$ compact的空间。重量大于或等于$ \ aleph_0 $的$％α$ -COMPACT空间是讨论了Cube $ d^{\ Aleph_0} $的$α$ - 连续图像。 $α$ - 连续函数的界限将映射到其他空间的$α$％compact空间。此外，$α$％ - 连续函数将$ x $映射到可计数的$α$ -compact Space $ y $是$α$ claped的子集$ x \ times y $。我们解释说，$α$ - 连续的功能将任何拓扑空间映射到$α$ compact的空间可以在某些限制下扩展到其域上。我们声称，$α$％compact的财产是$α$ - compact的，但相反的属性不是$ x $ $ x $的可计数联盟，也是$ x $的$α$ -compact。
In this paper, some features of countably $α$-compact topological spaces are presented and proven. The connection between countably $α$% -compact, Tychonoff, and $α$-Hausdorff spaces is explained. The space is countably $α$-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably $% α$-compact space with weight greater than or equal to $\aleph_0$ is the $α$-continuous image of a closed subspace of the cube $D^{\aleph_0}$ is discussed. The boundedness of $α$-continuous functions mapping $α$% -compact spaces to other spaces is cleared. Moreover, the $α$% -continuous function mapping the space $X$ to the countably $α$-compact space $Y$ is an $α$-closed subset of $X\times Y$ is argued and proved. We explained that the $α$-continuous functions mapping any topological space to a countably $α$-compact space can be extended over its domain under some constraints. We claimed that the property of being $α$% -compact is countably $α$-compact but the converse is not and the countable union of countably $α$-compact subspaces of $X$ is also countably $α$-compact.
