公理化局部有限空间中映射连续性的扩展

Extension of continuity of maps between axiomatic locally finite spaces

用公理化方法研究了局部有限空间中的连续映射及其扩张问题。给出了局部有限空间的公理化定义方法;利用邻近关系研究了局部有限空间中的连续映射、同胚和局部同胚等问题;通过对局部有限空间变形的研究,定义了局部有限空间的一种特殊收缩核,有效地解决了局部有限空间中连续映射的扩张问题。

The paper studies the extension problem of continuous maps between axiomatic locally finite spaces （for short, ALF spaces）. Indeed, an ALF space is a topological space satisfying a set of axioms suggested in [9, 10]. Further, an ALF space is defined by using a special kind of neighborhood different from the topological neighborhood in classical topology so that the con- tinuity of maps between ALF spaces can be defined by preserving the neighborhood relation （see Definition 10）. Therefore, it is necessary to develop the notions of continuity, homeomorphism and local homeomorphism for such spaces by using the neigh- borhood relation, which can be applicable in computer science. In the study of a deformation of an ALF space, we can develop a special kind of retract on ALF spaces. By using the retract, we can efficiently deal with the extension of continuous maps be- tween ALF spaces.