Survey of local connectedness axioms and their properties in [formula omitted]-topological spaces

Abstract

Based on the notions of fuzzy connectedness defined by Lu and Li [Fuzzy connectedness: new definitions and comparisons, Fuzzy Sets and Systems 157(2006)1928–1940], some new fuzzy local connectedness, such as local connectedness, local ultra- F 1 connectedness, local strong F 1 connectedness, local ultra- F 2 connectedness, local strong F 2 connectedness, ultra- F 1 local connectedness, strong F 1 local connectedness, ultra- F 2 local connectedness, and strong F 2 local connectedness, of L -topological spaces or L -cotopological spaces are defined in this paper. Apart from maintaining some frequently used properties that classical local connected topological spaces possess, L -topological spaces or L -cotopological spaces which satisfy our new fuzzy local connectedness axioms also have some interesting properties. For example, local connectedness, local ultra- F 1 connectedness, local ultra- F 2 connectedness, local strong F 1 connectedness, and local strong F 2 connectedness are all I ( L ) -inducible if L is a complete and meet-continuous deMorgan algebra; they are L -extension of local connectedness of topological spaces under some moderate restrictions to L . It is proved that all L -intervals are locally connected if Copr ( L ) is a ∨ -generating set of L . Some categorical results reflecting the differences between classical topology and L -topology are obtained, the relationships between these notions of connectedness are also investigated in detail.