Survey of local connectedness axioms and their properties in L-topological spaces

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-F1 connectedness, local strong F1 connectedness, local ultra-F2 connectedness, local strong F2 connectedness, ultra-F1 local connectedness, strong F1 local connectedness, ultra-F2 local connectedness, and strong F2 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-F1 connectedness, local ultra-F2 connectedness, local strong F1 connectedness, and local strong F2 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.