论文标题
$ p $ - 基础和拓扑组
$P$-bases and Topological Groups
论文作者
论文摘要
如果存在一个$ x $的拓扑空间$ x $，则在任何$ x $ in x $ in x $ in x $ p $的任何$ p $ base中，如果存在邻里base $（u_p [x]）_ {p \ in p} $ at $ x $，以便$ x $，以便$ x $，以便$ x $ $ u_p [x] \ subseteq u_ _ subseteq u_ {p'} $ $ $ p $ p $我们证明了一个紧凑的空间是可计数的，因此可以计算，如果它具有可计数的散射高度，并且是$ \ natercal {k}（m）$ - 一些可分开的度量公制空间$ m $的基础。这给出了\ cite {banakh2019}的问题8.6.8的积极答案。 令$ a（x）$为$ x $的免费Abelian拓扑组。结果表明，如果$ y $是$ x $的缩回，则免费的Abelian拓扑组$ a（y）$具有$ p $ - base，$ a（x/y）$具有$ q $ - base，则$ a（x）$具有$ p \ times q times q $ - basase。另外，如果$ y $是$ x $的封闭子空间，而$ a（x）$具有$ p $ bas，则$ a（x/y）$具有$ p $ - base。 结果表明，任何具有$ \ MATHCAL {K}（M）$的Fréche-urysohn拓扑组 - 对于某些可分开的度量空间$ m $的基础都是首先计算，因此可转换。而且，如果$ p $是具有口径〜$（ω_1，ω）$和$ g $的姿势，则是一个具有$ p $ bas的拓扑组，那么G中的任何前校准子集都是可分配的，因此$ g $是严格的天使般的。讨论了功能空间中的应用程序$ C_P（X）$和$ C_K（X）$。我们还举例说明了一个拓扑布尔的字符$ \ leq \ mathfrak {d} $的拓扑布尔，使得前事子集可迁移，但是$ g $如果$ω^ω$ - base如果$ω_1<\ m athfrak {d} $。这给出了\ cite {gkl15}中的问题6.5的一致的负面答案。
A topological space $X$ is defined to have a neighborhood $P$-base at any $x\in X$ from some poset $P$ if there exists a neighborhood base $(U_p[x])_{p\in P}$ at $x$ such that $U_p[x]\subseteq U_{p'}[x]$ for all $p\geq p'$ in $P$. We prove that a compact space is countable, hence metrizable, if it has countable scattered height and a $\mathcal{K}(M)$-base for some separable metric space $M$. This gives a positive answer to Problem 8.6.8 in \cite{Banakh2019}. Let $A(X)$ be the free Abelian topological group on $X$. It is shown that if $Y$ is a retract of $X$ such that the free Abelian topological group $A(Y)$ has a $P$-base and $A(X/Y)$ has a $Q$-base, then $A(X)$ has a $P\times Q$-base. Also if $Y$ is a closed subspace of $X$ and $A(X)$ has a $P$-base, then $A(X/Y)$ has a $P$-base. It is shown that any Fréche-Urysohn topological group with a $\mathcal{K}(M)$-base for some separable metric space $M$ is first-countable, hence metrizable. And if $P$ is a poset with calibre~$(ω_1, ω)$ and $G$ is a topological group with a $P$-base, then any precompact subset in G is metrizable, hence $G$ is strictly angelic. Applications in function spaces $C_p(X)$ and $C_k(X)$ are discussed. We also give an example of a topological Boolean group of character $\leq \mathfrak{d}$ such that the precompact subsets are metrizable but $G$ doesn't have an $ω^ω$-base if $ω_1<\mathfrak{d}$. This gives a consistent negative answer to Problem 6.5 in \cite{GKL15}.
