b_β-CONNECTEDNESS AS A SIMULTANEOUS CHARACTERIZATION OF β-CONNECTEDNESS, γ-CONNECTEDNESS AND β_β-CONNECTEDNESS

摘要

Two new type of sets called b_β-open and β_β-open sets are introduced in topological spaces. Based on these two new notions b_β-connectedness and β_β-connectedness are introduced, which are stronger than connectedness and equivalent to γ-connectedness and β-connectedness, respectively. Further, it is shown that b_β-connectedness is equivalent to β_β-connectedness. The properties of these notions are explored and their relationship with other forms of connectedness, for example hyperconnectedness, semi-connectedness, preconnectedness etc, are discussed. Some equivalent forms of locally indiscrete spaces are defined. The notions of locally b_β-connected and locally β_β-connected spaces are introduced. The behavior of b_β-connectedness and b_β-connectedness under several types of mappings is investigated. An intermediate value theorem is obtained.