On Topologies Generated by Lower Porosity

摘要

Abstract In the paper we study properties of a lower porosity of a set in a normed space (X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(X,Vert ;Vert )$$end{document}. Two topologies p̲(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${underline{p}}(X,Vert ;Vert )$$end{document} and s̲(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${underline{s}}(X,Vert ;Vert )$$end{document} on X generated by the lower porosity are defined. Relationships between these topologies and, previously defined by V. Kelar and L. Zajíček, topologies p(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p(X,Vert ;Vert )$$end{document} and s(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$s(X,Vert ;Vert )$$end{document} are studied. Applying topologies p̲(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${underline{p}}(X,Vert ;Vert )$$end{document} and s̲(X,‖‖)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${underline{s}}(X,Vert ;Vert )$$end{document} we characterize maximal additive class of lower porouscontinuous functions. Some relevant properties of defined topologies are considered.