通过软元素的视角构建软双拓扑空间
Pairwise Separation and Compactness in Soft Bitopological Spaces via Soft Elements
AI总结：
本文通过软元素视角构建软双拓扑空间，定义并比较了成对软分离公理与参数化公理，揭示了有限参数下软紧性的有限性现象。
摘要：
我们从软元素的角度引入软双拓扑空间。软双拓扑空间是一种配备了两个软拓扑的软集。按照Goldar-Ray的经典构造，每个在F上的软拓扑诱导出软元素集SE(F)上的普通拓扑；因此，每一个软双拓扑空间自然地确定了SE(F)上的真实双拓扑空间。在此设定下，我们定义了成对的软分离公理（T0，T1，T2）以及一种成对的软紧性概念，并将它们与参数化的对应概念进行比较。对于经典的（按截面生成的）软双拓扑，我们证明成对的软Ti公理等价于每个参数空间上对应的成对Ti公理。紧性表现出有限性现象：当参数集有限时，组件-wise的成对紧性强制成对的软紧性，而一个无限参数的例子表明有限性假设是必要的。例子用于说明诱导的双拓扑在SE(F)上可能与原始的软双拓扑行为不同。
英文摘要：
Let $F\colon A\to\mathcal P(X)$ be a soft set, and let $SE(F)$ be the set of all maps $a\colon A\to X$ such that $a(t)\in F(t)$, for every $t\in A$. If $τ$ is a soft topology on $F$, the sets $SE(H)$, where $H\inτ$, form a basis for an ordinary topology $τ^{\mathrm e}$ on $SE(F)$, called the \emph{induced soft-element topology}. Hence, two soft topologies on $F$ induce a bitopological space on $SE(F)$. We define pairwise soft $T_0$, $T_1$, and $T_2$ axioms using soft elements and prove that they are equivalent to the corresponding pairwise axioms of the induced soft-element bitopological space. For canonical soft bitopologies, these properties are also equivalent to the same axioms in every component bitopological space. We also study pairwise soft compactness. For canonical soft bitopological spaces, pairwise soft compactness implies pairwise compactness of every component space, and the converse holds when the parameter set is finite. An example with infinitely many parameters shows that the finiteness condition is needed.
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