This chapter deals with the K-analyticity of a topological space E and the concept of a resolution generated on E (i.e. a family of sets \(\{K_\alpha :\alpha \in \mathbb {N}^{\mathbb {N}}\}\) such that \({E}=\bigcup _{\alpha }{K}_{\alpha }\) and \(K_\alpha \subset K_\beta \ \mathrm {if}\ \alpha \leq \beta \) ). Compact resolutions (i.e. resolutions \(\{K_\alpha :\alpha \in \mathbb {N}^{\mathbb {N}}\}\) whose members are compact sets) naturally appear in many situations in topology and functional analysis. Any K-analytic space admits a compact resolution, and for many topological spaces X, the existence of such a resolution is enough for X to be K-analytic. Many of the ideas in the book are related to the concept of compact resolution. We gather some results, mostly due to Valdivia, about lcs’s admitting resolutions consisting of Banach discs and their relations with the closed graph theorems. We present Hurewicz and Alexandrov’s theorems as well as Calbrix–Hurewicz’s theorem, which yields that a regular analytic space X is not \(\sigma \) -compact if and only if X contains a closed subset homeomorphic to \(\mathbb {N}^{\mathbb {N}}\) .