On paracompactness in function spaces with the compact-open topology.
On paracompactness in function spaces with the compact-open topology.
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DOI：
10.1090/s0002-9939-1971-0276919-3
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发表时间：
1971
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影响因子：
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通讯作者：
P. O'Meara
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作者：
P. O'Meara
空间 X 的 k 网络 (P 是 X 的子集族，使得如果 C C U 具有 C 紧凑且 U 开放，则存在 (P 的成员的有限并集 R，使得 CCLRQ U。Ha 空间是具有可数 ¿ 网络的 r3 空间，it 空间是具有 <r 局部有限 ¿ 网络的 Tr 空间。在本文中，表明如果 X 是无空间且 F 是仿紧 N 空间，那么 G(X, Y)，具有紧开拓扑，是仿紧 N 空间。结果意味着，如果 X 是可分离度量，则 G(X, Y) 是仿紧空间。本文的目的是证明某些函数空间是仿紧空间，它具有 E. Michael 的无空间 [3] 的许多属性，但 N 空间不一定是 Lindelöf。无空间可以被表征为 Lindelöf 或遗传可分离的 N 空间 [7]，e(X, Y) 将表示配备紧开拓扑的从 X 到 Y 的连续函数的空间，Michael 已经证明，如果 X 和 Y 是无空间，则 Q(X, Y) 也是无空间。 &(X, Y) 是一个仿紧 N 空间。当删除“仿紧”时，不知道前面的陈述是否成立。 定义 1。一个 k 网络（空间 X 的 P 是 X 的子集族，如果 C C U 在 X 中是紧致的，U 是开放的，则存在 (P 的成员的有限并集 R，使得 CERE U。一个网络（空间 X 的 P 是 X 的子集族，如果 X EU 是开放的，则有a P£(P 使得 xEPEU。在 fe 网络的前述定义中，如果有限并集 R 被成员 P£(P 代替，则 (P 称为 X 的伪基)。E. Michael 和 A. Okuyama 分别在 [3] 和 [S] 中研究了具有可数伪基的正则 Pi 空间（即 N0 空间）和具有 c 局部有限网络的空间（即 <r 空间）。 ^-网络可能不是伪基，具有可数^-网络的空间也有可数伪基。编辑于 1970 年 4 月 2 日收到。AMS 1969 主题分类。
A k-network (P for a space X is a family of subsets of X such that if CC U, with C compact and U open, then there is a finite union R of members of (P such that CCLRQ U. An Ha-space is a r3-space having a countable ¿-network and an it-space is a Tr space having a <r-locally finite ¿-network. In this paper, it is shown that if X is an No-space and F is a paracompact N-space, then G(X, Y), with the compact-open topology is a paracompact N-space. The result implies that if X is separable metric and Y is metric, then G(X, Y) is paracompact. The purpose of this paper is to show that certain function spaces are paracompact. We shall introduce a class of spaces called N-spaces which share many of the properties of E. Michael's No-spaces [3], though an N-space need not be Lindelöf. The No-spaces can be characterized as Lindelöf or hereditarily separable N-spaces [7]. Throughout, e(X, Y) will denote the space of continuous functions from X to Y equipped with the compact-open topology. Michael has shown that if X and Y are No-spaces, then so is Q(X, Y). This theorem is generalized in Theorem 1, which states that if X is an N0-space and F is a paracompact N-space, then &(X, Y) is a paracompact N-space. It is unknown whether the preceding statement is true when "paracompact" is deleted. Definition 1. A k-network (P for a space X is a family of subsets of X such that if CC U, with C compact and U open in X, then there is a finite union R of members of (P such that CERE U. A network (P for a space X is a family of subsets of X such that if xEU, with U open, then there is a P£(P such that xEPEU. In the preceding definition of fe-network, if the finite union R is replaced by a member P£(P, then (P is called a pseudobase for X. Regular Pi-spaces with countable pseudobases (i.e., N0-spaces) and spaces with c-locally finite networks (i.e., <r-spaces) have been investigated by E. Michael and A. Okuyama in [3] and [S], respectively. Though a ^-network may not be a pseudobase, a space with a countable ^-network also has a countable pseudobase. Received by the editors April 2, 1970. AMS 1969 subject classifications. Primary 5428, 5450; Secondary 5435.