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The \(\sigma\)-property in \(C(X)\). (English) Zbl 1399.46032

Summary: The \(\sigma\)-property of a Riesz space (real vector lattice) \(B\) is: For each sequence \(\{b_{n}\}\) of positive elements of \(B\), there is a sequence \(\{\lambda_{n}\}\) of positive reals, and \(b\in B\), with \(\lambda_{n}b_{n}\leq b\) for each \(n\). This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “\(\sigma\)” obtains for a Riesz space of continuous real-valued functions \(C(X)\). A basic result is: For discrete \(X\), \(C(X)\) has \(\sigma\) iff the cardinal \(|X|<\mathfrak{b}\), Rothberger’s bounding number. Consequences and generalizations use the Lindelöf number \(L(X)\): For a \(P\)-space \(X\), if \(L(X)\leq\mathfrak{b}\), then \(C(X)\) has \(\sigma\). For paracompact \(X\), if \(C(X)\) has \(\sigma\), then \(L(X)\leq\mathfrak{b}\), and conversely if \(X\) is also locally compact. For metrizable \(X\), if \(C(X)\) has \(\sigma\), then \(X\) is locally compact.

MSC:

46E05 Lattices of continuous, differentiable or analytic functions
46A40 Ordered topological linear spaces, vector lattices
54C30 Real-valued functions in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D45 Local compactness, \(\sigma\)-compactness
54G10 \(P\)-spaces
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