Topological Space Function And Strong Extremely Connected Space

In this paper, we introduce some new notions in general topological spaces, general-ized topological spaces, and L-topological spaces, and then discuss their basic properties. In general topological spaces, we introduce notions of strongly semi-precontinuous function, B-closed space, strongly PΣspace, strongly regular space, strongly extremely disconnected space, and so on. In the strongly extremely disconnected space, compact space, B-closed space, S-closed space and H(i) space are all equivalent. In the generalized topological spaces, we introduce the concept of (αλ,λ')-continuity at a point, and discuss its basic properties and characterizations in detail. Then, we discuss the relationship between this function and other functions. In the L-topological spaces, we introduce and discuss the notion of Strong-ⅡNβ-compactness. The strong-ⅡNβ-compactness implies strong-ⅠNβ-compactness, hence it also implies Nβ-compactness,S*-compactness and Lowen's fuzzy compactness, but it is different from fuzzy semi-compactness. When L= [0,1], strong-ⅡNβ-compactness is equivalent to fuzzy semi-compactness.