Bounded resolutions for spaces Cp(X) and a characterization in terms of X

Abstract

An internal characterization of the Arkhangel’ski˘ı-Calbrixmain theorem from [4] is obtained by showing that the space Cp(X) of continuous real-valued functions on a Tychonoff space X is K-analytic framed in RX if and only if X admits a nice framing. This applies to show that a metrizable (or cosmic) space X is σ -compact if and only if X has a nice framing. We analyse a few concepts which are useful while studying nice framings. For example, a class of Tychonoff spaces X containing strictly Lindelöf Cˇ ech-complete spaces is introduced for which a variant of Arkhangel’ski˘ı-Calbrix theorem for σ-boundedness of X is shown.