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An internal characterization of the Arkhangel’skiĭ-Calbrix main theorem from [4] is obtained by showing that the space    $$C_{p}(X)$$  C p ( X ) of continuous real-valued functions on a Tychonoff space  X  is  K -analytic framed in    $$\mathbb {R}^{X}$$  R X if and only if  X  admits a  nice framing . This applies to show that a metrizable (or cosmic) space  X  is    $$\sigma $$  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 Čech-complete spaces is introduced for which a variant of Arkhangel’skiĭ-Calbrix theorem for    $$\sigma $$  σ -boundedness of  X  is shown.

Bounded resolutions for spaces $$C_{p}(X)$$ and a characterization in terms of X