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.


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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