Marciszewski Witold, Sobota Damian
Institute of Mathematics and Computer Science, University of Warsaw, Warsaw, Poland.
Department of Mathematics, Kurt Gödel Research Center, Vienna University, Vienna, Austria.
Arch Math Log. 2024;63(7-8):773-812. doi: 10.1007/s00153-024-00920-x. Epub 2024 Apr 9.
For a free filter on , endow the space , where , with the topology in which every element of is isolated whereas all open neighborhoods of are of the form for . Spaces of the form constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson-Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter , the space carries a sequence of normalized finitely supported signed measures such that for every bounded continuous real-valued function on if and only if , that is, the dual ideal is Katětov below the asymptotic density ideal . Consequently, we get that if , then: (1) if is a Tychonoff space and is homeomorphic to a subspace of , then the space of bounded continuous real-valued functions on contains a complemented copy of the space endowed with the pointwise topology, (2) if is a compact Hausdorff space and is homeomorphic to a subspace of , then the Banach space () of continuous real-valued functions on is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space contains a non-trivial convergent sequence, then the space () is not Grothendieck.
对于(\omega)上的一个自由滤子，赋予空间(X(\mathcal{F}))（其中(X(\mathcal{F})=\omega\cup{\infty})）一种拓扑，在该拓扑中(\omega)的每个元素都是孤立的，而(\infty)的所有开邻域都具有(A\cup{\infty})的形式，其中(A\in\mathcal{F})。形如(X(\mathcal{F}))的空间构成了最简单的非离散吉洪诺夫空间类。本文的目的是在巴拿赫空间理论中著名的约瑟夫森 - 尼森茨韦格定理的背景下研究它们。例如，我们证明，对于一个滤子(\mathcal{F})，空间(X(\mathcal{F}))承载一列归一化的有限支撑符号测度({\mu_n})，使得对于(X(\mathcal{F}))上的每个有界连续实值函数(f)，(\lim_{n\rightarrow\infty}\mu_n(f)=0)当且仅当(\mathcal{F})，即对偶理想(\mathcal{F}^*)在渐近密度理想(\mathcal{Z})之下是卡捷托夫的。因此，我们得到如果(\mathcal{F})，那么：(1) 如果(Y)是一个吉洪诺夫空间且(Y)同胚于(X(\mathcal{F}))的一个子空间，那么(Y)上有界连续实值函数的空间(C_b(Y))包含赋予点态拓扑的空间(C_p(X(\mathcal{F})))的一个补空间，(2) 如果(K)是一个紧致豪斯多夫空间且(K)同胚于(X(\mathcal{F}))的一个子空间，那么(K)上连续实值函数的巴拿赫空间(C(K))不是一个格罗滕迪克空间。后一个结果推广了一个众所周知的事实，即如果一个紧致豪斯多夫空间(K)包含一个非平凡收敛序列，那么空间(C(K))不是格罗滕迪克空间。
