论文标题
通用的核心可计可数第二个空间
A universal coregular countable second-countable space
论文作者
论文摘要
A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $ x \ setMinus（\ bar u_1 \ cap \ dots \ cap \ bar u_n）$是常规的拓扑空间）。核格超级连接空间的规范示例是投射空间$ \ mathbb q \ Mathsf p^\ p^\ infty $的拓扑矢量空间$ \ mathbb q^{<ω} = \ {（x_n） 0 \} | <ω\} $在理性的字段上$ \ mathbb q $。空间$ \ MATHBB Q \ MATHSF P^\ infty $是$ \ Mathbb Q^{<ω} \ setMinus \ {0 \}^ω$的商品空间，均等$ x \ sim y $ y $ sim y $ iff $ \ sim y $ iff $ \ mathbb q q { 我们证明，每个可计数的第二个可数的核心空间都是同质的，对于$ \ mathbb q \ mathsf p^\ p^\ iftty $，拓扑空间$ x $是同源的，对$ \ mathbb q \ mathbb q \ mathsf p^\ infty $ if且仅在$ x $ ables of x $ blecters and sequence and sequence and sequence and sequence and Adm sequine a d of secouts and phsect an $（x_n）_ {n \inΩ} $这样，以至于（i）$ x_0 = x $，$ \ bigcap_ {n \inΩ} x_n = \ emptySet $，（ii）对于每个$ n \inΩ$和一个非公开式开放式$ \ u \ seet u \ subsete u \ subsete u \ subseteq x_n $ yii $ un $ n \inΩ$补充$ x \ setminus x_n $是常规的拓扑空间。使用$ \ Mathbb Q \ Mathsf P^\ infty $的拓扑表征，我们找到了空间的拓扑副本$ \ Mathbb Q \ Mathsf P^\ infty $在商的空间，集团操作的轨道空间以及可计算拓扑领域的拓扑矢量空间的投射空间中。
A Hausdorff topological space $X$ is called $\textit{superconnected}$ (resp. $\textit{coregular}$) if for any nonempty open sets $U_1,\dots U_n\subseteq X$, the intersection of their closures $\bar U_1\cap\dots\cap\bar U_n$ is not empty (resp. the complement $X\setminus (\bar U_1\cap\dots\cap\bar U_n)$ is a regular topological space). A canonical example of a coregular superconnected space is the projective space $\mathbb Q\mathsf P^\infty$ of the topological vector space $\mathbb Q^{<ω}=\{(x_n)_{n\inω}\in \mathbb Q^ω:|\{n\inω:x_n\ne 0\}|<ω\}$ over the field of rationals $\mathbb Q$. The space $\mathbb Q\mathsf P^\infty$ is the quotient space of $\mathbb Q^{<ω}\setminus\{0\}^ω$ by the equivalence relation $x\sim y$ iff $\mathbb Q{\cdot}x=\mathbb Q{\cdot}y$. We prove that every countable second-countable coregular space is homeomorphic to a subspace of $\mathbb Q\mathsf P^\infty$, and a topological space $X$ is homeomorphic to $\mathbb Q\mathsf P^\infty$ if and only if $X$ is countable, second-countable, and admits a decreasing sequence of closed sets $(X_n)_{n\inω}$ such that (i) $X_0=X$, $\bigcap_{n\inω}X_n=\emptyset$, (ii) for every $n\inω$ and a nonempty open set $U\subseteq X_n$ the closure $\bar U$ contains some set $X_m$, and (iii) for every $n\inω$ the complement $X\setminus X_n$ is a regular topological space. Using this topological characterization of $\mathbb Q\mathsf P^\infty$ we find topological copies of the space $\mathbb Q\mathsf P^\infty$ among quotient spaces, orbit spaces of group actions, and projective spaces of topological vector spaces over countable topological fields.
