An Exploration About Generalized Open Sets And Generalized Continuity

Since A.Csaszar introduced the notion of generalized topology in 2002, many scholars have established the theories of basic sets, continuity, connectedness, separations, product and covering properties, which complete the theory of generalized topology. The theories of generalized open sets and continuity, as the basic concept of point-set topology theory, naturally attract many interests. Based on the preciously existing results, this article is mainly to study two subjects mentioned above in generalized topological spaces.1. Based on the notions of four special open sets provided by A.Csaszar, for five generalized open sets about the property of strong generalized topological space, we have(X, g) 、(X,a(g))、(X,p(g)) have the consensus on the property of strong generalized topological space. What’s more,(X,s(g)) 与(X,b(g)) are always strong generalized topological space. As an application of the conclusions, the extremely disconnected generalized topological space is a strict generalization of the strong topological space. Furthermore, the paper investigates some properties between the extremely disconnected generalized topological space and the disconnected generalized topological space.2. As an analogy of(ψ,ψ’)-continuity and weak(ψ,ψ’)-continuity, using the methods of researching almost(g, g’)-continuity, this paper introduces the notion of almost(ψ,ψ’)-continuity. Then, It provides the characterizations for this mapping. In addition, we investigate the relationship between almost(ψ,ψ’)-continuity and almost(g, g)y y ￠-continuity(respectively weak(ψ,ψ’)- continuity,(ψ,ψ’)- continuity).3. It is provided with some results about the relationship of(g, g’)- continuity, weak(g, g’)- continuity and almost(g, g’)- continuous mapping which are based on W.K.Min’s research. Firstly, some properties of g-open mappings, g-regular spaces and frontier of a set in generalized topological space are given. Secondly, with the help of these results, we have some conclusions: If weak(g, g’)- continuous mapping is g-open mapping, it’s also almost(g, g’)- continuous mapping. If the image space is g-regular space, weak(g, g’)- continuous mapping is(g, g’)- continuous mapping. If the preimage space is quasi generalized topological space, and the preimage of the frontier of every open set in the image space is a closed set, weak(g, g’)- continuous mapping is(g, g’)- continuous mapping.