Weakly monotone sets and continuous selection from a near-best approximation operator

A new notion of weak monotonicity of sets is introduced, and it is shown that an approximatively compact and weakly monotone connected (weakly Menger-connected) set in a Banach space admits a continuous additive (multiplicative) $\varepsilon $-selection for any $\varepsilon >0$. Then a notion of weak monotone connectedness (weak Menger connectedness) of sets with respect to a set of $d$-defining functionals is introduced. For such sets, continuous $(d^{-1},\varepsilon )$-selections are constructed on arbitrary compact sets.