A SEQUENTIAL PROPERTY OF Cp(X) AND A COVERING PROPERTY OF HUREWICZ
A SEQUENTIAL PROPERTY OF Cp(X) AND A COVERING PROPERTY OF HUREWICZ
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Cp(X)的序列性质和HUREWICZ的覆盖性质
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发表时间：
1997
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通讯作者：
M. Scheepers
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作者：
M. Scheepers
Cp（X）具有单调序列选择性质，如果对于每个f，对于每个序列（σn：n < ω），其中对于每个n σn是逐点单调收敛到f的序列，存在一个序列（fn：n < ω），使得对于每个n fn是σn的一项，并且（fn：n < ω）逐点收敛到f。本文证明了一个定理，它蕴涵了度量空间X的Cp（X）具有单调序列选择性质当且仅当X具有Hurewicz覆盖性质。证明了函数空间的闭包算子的性质与定义域空间的开覆盖性质之间的对偶关系。为了列出一些，让我们首先修复约定和符号。贯穿X的是一个拓扑空间，它至少具有吉洪诺夫分离性质-这意味着X的每一个元素子集都是闭的，并且每当C是X的闭子集并且x是来自C的补的点时，则存在从X到闭单位区间I的连续函数，该函数将x映射到0，并将C的每个元素映射到1。从X到真实的直线R的所有函数的集合，记为R，被认为是真实的直线的幂，并被赋予吉洪诺夫积拓扑。从X到R的连续函数集是R的子集;当赋予它从R继承的拓扑时，这个空间记为Cp（X）。Cp（X）的拓扑称为逐点收敛拓扑。从X到R的所有常数函数都是Cp（X）的元素。处处等于零的函数记为o。Cp（X）的闭包算子可能相当复杂。一个空间被称为具有可数紧性，如果对于每个子集A，一个点在A的闭包中当且仅当它在A的可数子集的闭包中。Arkhangel′ skelyi和Pytkeev的一个著名定理暗示Cp（X）具有可数紧性当且仅当X的每个有限方幂具有Lindelöf性质（每个开覆盖具有可数子覆盖）。Gerlits和Nagy发现了另一个有趣的等价物：一个空间的开覆盖是一个ω-覆盖，如果这个空间本身不是一个成员。1991年数学学科分类。小学54 E99.
Cp(X) has the monotonic sequence selection property if there is for each f , and for every sequence (σn : n < ω) where for each n σn is a sequence converging pointwise monotonically to f , a sequence (fn : n < ω) such that for each n fn is a term of σn, and (fn : n < ω) converges pointwise to f . We prove a theorem which implies for metric spaces X that Cp(X) has the monotonic sequence selection property if, and only if, X has a covering property of Hurewicz. Some nice duality results have been proved which relate properties of the closure operator of function spaces with open covering properties of the domain spaces. To list a few, let us first fix conventions and notations. Throughout X will be a topological space which has at least the Tychonoff separation property – this means that every one element subset of X is closed, and whenever C is a closed subset of X and x is a point from the complement of C, then there is a continuous function from X to the closed unit interval I which maps x to 0 and maps each element of C to 1. The set of all functions from X to the real line R, denoted R , is considered as a power of the real line and is endowed with the Tychonoff product topology. The set of continuous functions from X to R is a subset of R ; when endowed with the topology it inherits from R , this space is denoted Cp(X). The topology of Cp(X) is known as the topology of pointwise convergence. All constant functions from X to R are elements of Cp(X). The function which is everywhere equal to zero is denoted o. The closure operator of Cp(X) can be quite complicated. A space is said to have countable tightness if for every subset A, a point is in the closure of A if, and only if, it is in the closure of a countable subset of A. A well– known theorem of Arkhangel′skǐi and Pytkeev implies that Cp(X) has countable tightness if, and only if, every finite power of X has the Lindelöf property (every open cover has a countable subcover). Gerlits and Nagy found another interesting equivalent: An open cover of a space is an ω–cover if the space itself is not a member Received by the editors December 15, 1995 and, in revised form, April 11, 1996. 1991 Mathematics Subject Classification. Primary 54E99.