Plausible Reasoning Of SOme Topological Theorems

Abstract Plausible reasoning is an important means for learning, teach-ing, and studying mathematics. It's researched in elementary mathematics mainly, but less in Higher Mathematics. This article gives an example of the applications of plausible reasoning, which is the learning, teaching, and researching of Higher Mathematics. Through "plausible" inducting and analogying, this paper shows that how to initially establish the theory of connectedness and local connectedness of pre-topological spaces, which is the generalization of topological spaces. All of these are found because of those related conclusions about connectedness and local connectedness of topological spaces in the college mathematics textbooks.The main points of this paper are as follows:In chapter one, the application of plausible reasoning in the learning and teach-ing of Calculus is discussed. Induction and analogy of plausible reasonable are intro-duced firsrly, and then some all-known theories in Calculus are chosen, for example, the square sum of the inverse of natural numbers and the auxiliary function in La-grange's mean value theorem. According to some information and the understanding of myself the interesting discovery process of these theories are reproducted, and the importance of reproducting this process consciously in the learning and teaching of Calculus is given.In chapter two, in the learning and teaching of Point Set Topology, the examples of the application of plausible reasoning is given. It main introduces some conclu-sions being published which are discovered through using plausible reasoning con-sciously when studying Piont Set Topology in the university. The basic concepts of pre-topological space are given firstly, including continuous mapping, product space, connected space, strong connectedness, and local strong connectedness space. On this basis, the discovery process of some typical properties about strong connected-ness and local strong connectedness are given in detail, for example, the equivalent characterization of strong connectedness, the property unchanged under continuous mappings, some properties of strongly connected subsets, the properties that if the spaces are local strong connectedness, then their open sebsets. sums, products, quo-tients are all local strong connectedness. Plausible reasoning is consciously applied in the studying and teaching of Topology. In chapter three, the risk of plausible reasoning is discussed. That plausible reasonging has some risk is introduced firstly, for example, when calculating the sum of the infinite series, the order of its items can't be changed, and we should strictly follow the concepts. Then according to the experience of myself in the university, this paper shows that only to owning the "feeling" to mathematics, plausible reasoning is possible.