Sequences of semicontinuous functions accompanying continuous functions
Sequences of semicontinuous functions accompanying continuous functions
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DOI：
10.1016/j.topol.2009.07.017
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发表时间：
2009-11
影响因子：
0.6
通讯作者：
H. Ohta;M. Sakai
中科院分区：
文献类型：
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作者：
H. Ohta;M. Sakai
一个空间X被称为具有性质（USC）（分别是）。（LSC）），如果{fn：n∈ω}是一个上序列（分别为下）连续函数从X到闭单位区间[0，1]逐点收敛到值为0的常数函数0，存在一列{gn：n∈ω}从X到[0，1]的连续函数，使得fn <$gn（n∈ω）和{gn：n∈ω}逐点收敛到0。在本文中，我们研究具有这些性质和相关的空间。特别地，我们证明了：（a）对于真实的直线的子集X，X具有（USC）性质当且仅当它是σ-集;（B）如果X是不可测基数空间，且具有（LSC）性质，则它是离散的。我们的研究来源于Scheepers关于性质S1（Γ，Γ）和wQN的猜想。
A space X is said to have property (USC) (resp. (LSC)) if whenever {fn:n∈ω} is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [0,1] converging pointwise to the constant function 0 with the value 0, there is a sequence {gn:n∈ω} of continuous functions from X into [0,1] such that fn⩽gn(n∈ω) and {gn:n∈ω} converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S1(Γ,Γ) and wQN.