论文标题
本地封闭的集合和次最大空间
Locally closed sets and submaximal spaces
论文作者
论文摘要
如果$ x $的每个密集子集都打开，则拓扑空间$ x $称为sumpaximal。在本文中，我们表明，如果$βx$，$ x $的石icech紧凑型是一个次最大的空间，那么$ x $是一个紧凑的空间，因此$βx= x $。我们还证明，如果$ \ upsilon x $（$ x $的hewitt realmpactification''是次最大的且首先可计数，而$ x $没有孤立的点，那么$ x $是realcompact，因此$ \ upsilon x = x $。我们观察到，每个次最大的Hausdorff空间都是伪限的。事实证明，如果$ \ upsilon x $是一个次最大的空间，那么$ x $是伪金融$μ$ $ compact空间。 给出了一个示例，该示例表明$ x $可能是超大的，但是$ \ upsilon x $可能不是相互临时的。给定拓扑空间$（x，{\ Mathcal t}）$，$ x $的所有本地封闭子集的集合形成了$ x $上的拓扑基础，这是$ {\ Mathcal t_l} $表示的。我们研究$（x，{\ Mathcal t}）$和$（x，{\ Mathcal t_l}）$之间的一些拓扑属性，例如我们表明a）$（x，x，{\ nathcal t_l}）$是离散的。 b）$（x，{\ mathcal t}）$是且仅当$ {\ Mathcal t} = {\ Mathcal t_l} $时，仅当$ {\ mathcal t} =; c）$（x，{\ mathcal t}）$是且仅当连接$（X，{\ Mathcal t_l}）$时。我们看到，在本地不差的空间中，$ t_0 $，$ t_d $，$ t_ \ frac {1} {1} {2} $，$ t_1 $，sumpagimal and Invide Conconce的概念。最后，我们证明了LC-CORCACT空间的每个Clopen子空间都是LC-Compact。
A topological space $X$ is called submaximal if every dense subset of $X$ is open. In this paper, we show that if $βX$, the Stone-Čech compactification of $X$, is a submaximal space, then $X$ is a compact space and hence $βX=X$. We also prove that if $\upsilon X$, the Hewitt realcompactification of $X$, is submaximal and first countable and $X$ is without isolated point, then $X$ is realcompact and hence $\upsilon X=X$. We observe that every submaximal Hausdorff space is pseudo-finite. It turns out that if $\upsilon X$ is a submaximal space, then $X$ is a pseudo-finite $μ$-compact space. An example is given which shows that $X$ may be submaximal but $\upsilon X$ may not be submaximal. Given a topological space $(X,{\mathcal T})$, the collection of all locally closed subsets of $X$ forms a base for a topology on $X$ which is denotes by ${\mathcal T_l}$. We study some topological properties between $(X,{\mathcal T})$ and $(X,{\mathcal T_l})$, such as we show that a) $(X,{\mathcal T_l})$ is discrete if and only if $(X,{\mathcal T})$ is a $T_D$-space; b) $(X,{\mathcal T})$ is a locally indiscrete space if and only if ${\mathcal T}={\mathcal T_l}$; c) $(X,{\mathcal T})$ is indiscrete space if and only if $(X,{\mathcal T_l})$ is connected. We see that, in locally indiscrete spaces, the concepts of $T_0$, $T_D$, $T_\frac{1}{2}$, $T_1$, submaximal and discrete coincide. Finally, we prove that every clopen subspace of an lc-compact space is lc-compact.
