Upper Porous Sets which are Not-σ-Lower Porous

Abstract

Let $X$ be a nonempty, topologically complete metric space with no isolated points. We show that there exists a closed upper porous set (in~a~strong sense) $F\subset X$ which is not $\s$-lower porous (in a weak sense). More precisely, we show that there exists a closed $(g_1)$-shell porous set $F\subset X$ which is not $\s$-$(g_2)$-lower porous, where $g_1$ and~$g_2$ are arbitrary admissible functions.