The authors prove several results related to the following problem: Is every \(\sigma\)-compact space with the compact extension property an ANR? This problem appears in the list of ‘Open problems in infinite dimensional topology’ collected by J. E. West in [Open problems in topology (ed. by J. van Mill and G. M. Reed) (1990; Zbl 0718.54001), pp. 523-597]. The main result proved by the authors is a characterization of \(\sigma\)-compact ANR’s. They prove that a \(\sigma\)- compact space is an ANR if and only if it is locally equi-connected and has the compact neighborhood extension property. An important element in the proof is the useful characterization of ANR’s given by the first author in [Fundam. Math. 124, 243-254 (1984; Zbl 0573.54009)]. The authors derive some interesting corollaries, among which we mention the following: Let \(X\) be the union of an increasing family of ANR’s \(X_ 1\subset X_ 2\subset\dots\) closed in \(X\). Then \(X\) is an ANR if \(X\) is locally equi-connected.