Contreras-Carreto, A.; Tamariz-Mascarúa, A. On some generalizations of compactness in spaces \(C_p(X,2)\) and \(C_p(X,\mathbb Z)\). (English) Zbl 1042.54010 Bol. Soc. Mat. Mex., III. Ser. 9, No. 2, 291-308 (2003); erratum ibid. 10, No. 1, 167-168 (2004). The authors discuss duality between properties of a Tychonoff space \(X\) and several generalizations of compactness of spaces \(C_p(X,\mathbb Z)\), \(C_p(X,{\mathbf 2})\) and \(C_p(X,{\mathbf n})\), \(n\in \mathbb N\). The results are compared with the corresponding known facts concerning the space \(C_p(X)\) of continuous real-valued functions on \(X\) with the pointwise topology. Two main results: (1) a zero-dimensional space \(X\) is an Eberlein compactum iff \(C_p(X,\mathbb Z)\) is \(\sigma\)-compact, (2) for a normal zero-dimensional space \(X\), \(C_p(X,{\mathbf 2})\) is \(\sigma\)-compact iff \(X\) is an Eberlein-Grothendieck compactum and the set \(X'\) of non-isolated points of \(X\) is an Eberlein compact space. Reviewer: Ljubiša D. Kočinac (Aleksandrovać) Cited in 1 ReviewCited in 6 Documents MSC: 54C35 Function spaces in general topology 54D45 Local compactness, \(\sigma\)-compactness 54C50 Topology of special sets defined by functions 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) Keywords:function spaces; \(C_\alpha\)-compact; \(\alpha\)-pseudocompact; ultracompact; pseudocompact; Eberlein-Grothendieck space; Eberlein compact PDFBibTeX XMLCite \textit{A. Contreras-Carreto} and \textit{A. Tamariz-Mascarúa}, Bol. Soc. Mat. Mex., III. Ser. 9, No. 2, 291--308 (2004; Zbl 1042.54010)