Properties of Sets in Asymmetric Spaces

For a space \(X=(X,||\,{\cdot}\,|)\) with asymmetric seminorm, the asymmetric closure seminorm \(\overline{||\,{\cdot}\,|}\) is defined as the Minkowski functional of the \(||\,{\cdot}\,|\) -closure of the ball \(B(0,1)\) . It is shown that an asymmetric normed space is Hausdorff if and only if the closure seminorm is an asymmetric norm on this space. It is also shown that \(||x||_{\Sigma}\leq\overline{||x|}\) , where \(||x||_{\Sigma}:=\inf_{y\in X}(||y|+||y-x|)\) . Approximatively compact sets in asymmetric normed \((\textrm{CLUR})\) -spaces \((X,\overline{||\,{\cdot}\,|})\) are studied.