arXiv 1606.010132016-06-03math.GNmath.FA

函数空间的Ascoli性质

The Ascoli property for function spaces

Saak Gabriyelyan, Jan Grebík, Jerzy Kakol, Lyubomyr Zdomskyy

备注：: 15 pages. Comments are welcome

摘要：

本文研究了Tychonoff空间$X$上的Ascoli空间$C_p(X)$和$C_k(X)$.阿斯科利空间类$X$，即$C_k(X)$的任何紧子集$K$是均匀连续的空间$X$，本质上包括$k_{\mathbb R}$ -空间类。首先证明了如果$C_p(X)$是Ascoli，则它是$κ$ -Fréchet-Urysohn.如果$X$是宇宙的，那么$C_p(X)$是Ascoli当且仅当它是$κ$ -Fr'echet-Urysohn。这导致以下扩展的结果Morishita：如果一个Čech完全空间$X$的空间$C_p(X)$是Ascoli，那么$X$是分散的。如果$X$是离散的和可分层的，则$C_p(X)$是Ascoli空间。因此：（a）如果$X$是完备可度量化空间，则$C_p(X)$是Ascoli当且仅当$X$是离散的。(b)若$X$是Čech完备Lindelöf空间，则$C_p(X)$是Ascoli当且仅当$X$是离散的当且仅当$C_p(X)$是Fréchet-Urysohn。证明了对点可数型仿紧空间$X$，下列条件等价：（i）$X$是局部紧的. (ii)$C_k(X)$是一个$k_{\mathbb R}$ -空间。(iii)$C_k(X)$是一个Ascoli空间。研究了Asoli空间$C_k(X,[0,1])$.

英文摘要：

The paper deals with Ascoli spaces $C_p(X)$ and $C_k(X)$ over Tychonoff spaces $X$. The class of Ascoli spaces $X$, i.e. spaces $X$ for which any compact subset $K$ of $C_k(X)$ is evenly continuous, essentially includes the class of $k_{\mathbb R}$-spaces. First we prove that if $C_p(X)$ is Ascoli, then it is $κ$-Fréchet-Urysohn. If $X$ is cosmic, then $C_p(X)$ is Ascoli iff it is $κ$-Fr'echet-Urysohn. This leads to the following extension of a result of Morishita: If for a Čech-complete space $X$ the space $C_p(X)$ is Ascoli, then $X$ is scattered. If $X$ is scattered and stratifiable, then $C_p(X)$ is an Ascoli space. Consequently: (a) If $X$ is a complete metrizable space, then $C_p(X)$ is Ascoli iff $X$ is scattered. (b) If $X$ is a Čech-complete Lindelöf space, then $C_p(X)$ is Ascoli iff $X$ is scattered iff $C_p(X)$ is Fréchet-Urysohn. Moreover, we prove that for a paracompact space $X$ of point-countable type the following conditions are equivalent: (i) $X$ is locally compact. (ii) $C_k(X)$ is a $k_{\mathbb R}$-space. (iii) $C_k(X)$ is an Ascoli space. The Asoli spaces $C_k(X,[0,1])$ are also studied.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

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