拓扑空间中的三个典型反例

Three Typical Counter-examples in the Topological Space

拓扑空间中的反例,在学习和研究拓扑学理论中起着重要的作用,一个好的反例可以为拓扑理论找出存在的依据。这里给出三个反例,存在两个度量空间X与Y,使X2与Y2等距而X与Y并不等距是拓扑空间中的反例;存在不可度量化的紧的完全正规空间是拓扑空间分离性的反例;不存在非零连续线性泛函的线性拓扑空间是线性拓扑空间的反例。

Counter-examples in the topological space play an important role in the topology learning and research.A good counter-example can find out the basis for the existence of topological theory.Three counter-examples of topological space are proposed in this study： There are two metric spaces,X and Y,in which and are isometric,whileand are not isometric;Non-metric tight regular space is a counter-example of the separation of topological space;No non-zero continuous linear functionals of the linear topological space is a counter-example of linear topological space.