Concerning rings of continuous functions
Concerning rings of continuous functions
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关于连续函数环
DOI：
10.1090/s0002-9947-1954-0063646-5
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发表时间：
1954
影响因子：
1.3
通讯作者：
M. Henriksen
中科院分区：
文献类型：
--
作者：
L. Gillman;M. Henriksen
本文研究了完全正则拓扑空间X上连续实值函数环C（X，R）的两个不同但相关的问题。第一种，治疗？? 1-7，是研究我们称之为P-空间的那些空间X，使得环C（X，R）的每个素理想都是极大理想。这个问题的背景和动机是在？1.这些结果包含了关于环C（X，R）的素理想的一系列定理，特别是P-空间的一系列刻画。第二个问题，在讨论？？8-10，是对休伊特所称的Q-空间的研究-这些空间X不能嵌入任何更大的空间的稠密子集，C（X，R）中的每个函数都可以连续扩展。这个问题的介绍是提供？8.我们讨论的Q-空间是有限的类线性有序空间（介绍？6）。我们能够解决一个问题，当一个任意的线性序空间是或不是一个Q-空间。仿紧空间的概念原来与这些考虑密切相关。我们还得到了线性序仿紧空间的一个特征，特别地，我们发现每个线性序Q-空间都是仿紧的。沿着得到的一个结果是：每个线性序空间都是可数仿紧的。
The present paper deals with two distinct, though related, questions, concerning the ring C(X, R) of all continuous real-valued functions on a completely regular topological space X. The first of these, treated in ? ?1-7, is the study of what we call P-spacesthose spaces X such that every prime ideal of the ring C(X, R) is a maximal ideal. The background and motivation for this problem are set forth in ?1. The results consist of a number of theorems concerning prime ideals of the ring C(X, R) in general, as well as a series of characterizations of P-spaces in particular. The second problem, discussed in ??8-10, is an investigation of what Hewitt has termed Q-spaces-those spaces X that cannot be imbedded as a dense subset of any larger space over which every function in C(X, R) can be continuously extended. An introduction to this question is furnished in ?8. Our discussion of Q-spaces is confined to the class of linearly ordered spaces (introduced in ?6). We are able to settle the question as to when an arbitrary linearly ordered space is or is not a Q-space. The concept of a paracompact space turns out to be intimately related to these considerations. We also derive a characterization of linearly ordered paracompact spaces, and we find in particular that every linearly ordered Q-space is paracompact. A result obtained along the way is that every linearly ordered space is countably paracompact.