介于点态拓扑与一致收敛拓扑之间的C(X)上的一类特殊拓扑
A special kind of topology on $C(X)$ lying between the point-open topology and the topology of uniform convergence
- University of Calcutta（加尔各答大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对拓扑空间X定义了一类介于点态拓扑与一致收敛拓扑之间的C(X)上的特殊拓扑，揭示其连通性、可度量化性等性质与X的伪紧性、ℵ_α-可分性的等价关系，引入两个新基数函数并证明其对应C_{ℵ_α}(X)的特征与伪特征，还得到相关基数函数相等的结论。
AI中文摘要：
设X为拓扑空间，对每个超限基数ℵ_α，我们在环C(X)上定义拓扑C_{ℵ_α}(X)；当ℵ_α=ℵ_0时，C_{ℵ_α}(X)退化为空间C_p(X)。我们证明：当ℵ_α≥ℵ_1时，C_{ℵ_α}(X)是道路连通的当且仅当它是连通的当且仅当X是伪紧的。此处我们定义了ℵ_α-可分空间，进一步证明X是ℵ_α-可分的当且仅当C_{ℵ_α}(X)是可度量化的当且仅当它是序列空间。随后我们引入两个新基数函数cc_{ℵ_α}(X)和ac_{ℵ_α}(X)，它们分别是C_{ℵ_α}(X)的特征和伪特征。文末我们证明与空间C_{ℵ_α}(X)相关的若干基数函数彼此相等。
英文摘要：
Let $X$ be a topological space. For each transfinite cardinal number $\aleph_α$, we define a topology $C_{\aleph_α}(X)$ on the ring $C(X)$. With $\aleph_α=\aleph_0$, $C_{\aleph_α}(X)$ reduces to the space $C_p(X)$. We prove that for $\aleph_α\geq \aleph_1$, $C_{\aleph_α}(X)$ is pathwise connected if and only if it is connected if and only if $X$ is pseudocompact. Here, we define $\aleph_α$-separable space and furthermore, we show that $X$ is $\aleph_α$-separable when and only when $C_{\aleph_α}(X)$ is metrizable when and only when it is a sequential space. Later we introduce two new cardinal functions, namely $cc_{\aleph_α}(X)$ and $ac_{\aleph_α}(X)$ which turned out to be the character and pseudocharacter of $C_{\aleph_α}(X)$ respectively. At the end of this article we show that a number cardinal functions associated with the space $C_{\aleph_α}(X)$ are equal.
