On mappings with strongly semi-closed graphs

Abstract

우리는 strongly 반폐그래프라는, 새로운 약화된 폐그래프를 정의하여, 그 성질과 이미 알려진 여러 약화된 폐그래프들의 성질과 비교하고, 새로 도입된 이 개념은 폐그래프와는 독립적임을 보인다.

여러 사상들이 Strongly 반폐그래프를 가질 때 성질을 알아보고 그 개념을 이용 Semi-T₁공간, semi-T₂공간들의 성질을 알아본다.

A mapping f : X→Y has a strongly semi-closed graph iff for each(x,y) G(f), there exist semi-open sets U and V containing x, y, respectively, such that (U×scl V)∩G(f)= . This type of graph is weaker than a strongly-closed graph [12] and is stronger than a semi-closed graph [7]. However, it is independent of a closed graph. Several known mappings may fail to have a strongly semi-closed graph and similarly, a mapping having a strongly semi-closed graph need not be a certain type of mappings. Conditions under which a certain mapping has a strongly semi-closed graph and conditions under which range or/and domain of a mapping having a strongly semi-closed graph become semi-T₁or/and semi-T₂are obtained.

A mapping f : X→Y has a strongly semi-closed graph iff for each(x,y) G(f), there exist semi-open sets U and V containing x, y, respectively, such that (U×scl V)∩G(f)= . This type of graph is weaker than a strongly-closed graph [12] and is stronger than a semi-closed graph [7]. However, it is independent of a closed graph. Several known mappings may fail to have a strongly semi-closed graph and similarly, a mapping having a strongly semi-closed graph need not be a certain type of mappings. Conditions under which a certain mapping has a strongly semi-closed graph and conditions under which range or/and domain of a mapping having a strongly semi-closed graph become semi-T₁or/and semi-T₂are obtained.