拓扑空间公理的教学困难成因与基于收敛“直观”的对策

The Reason of Teaching Difficulties on Axioms of Topological Spaces and Countermeasure Based on Convergence’s Intuition

指出分析学背景的欠缺,拓扑学所使用的形式公理化方法以及学生对公理化方法的陌生,导致了拓扑空间公理的教学困难.针对拓扑空间开(闭)集公理的引入教学,对策是在实直线上,从关于数列极限运算封闭的“直观”属性出发,用开区间和闭区间对数列极限运算封闭的对偶性,分别定义实数集上的开集与闭集,则二者自然地呈现对偶性,利于理解开集的“有限交与任意并封闭”公理,以期缓解拓扑空间公理的教学困难.

It points out the lack of analytical background, the formal axiomatic methods used in topology, and students’unfamiliarity with axiomatic methods, leads to the teaching difficulties on axioms of topological spaces. On teaching open(closed) sets’s axioms in topological spaces, solution is inspired by "intuitive" property of sequence limits’closedness of open and closed intervals on real numbers. Then open sets and closed sets on real numbers are respectively defined, and they are naturally dual. Consequently, it would be helpful to understand open set axiom of "closedness to arbitrary union and finite intersection" and relieve teaching difficulties on the axiom of topological spaces.