| The problem of topologise counting was put forward in the1930s. And whatmost mathematicians have been studying is to find the equation between the numberof elements of a finite set and the quantity of topologies on it, but failed. Currently,most methods is to enumerate, and specifically for a set containing n elements thetime complexity, if to enumerate, is as great as22n, increasing the difficulty of theproblem.Studies, based on recursive thought, about generating topologies on a finite set isdone in the article and three algorithms,different from the traditional enumeration,were put forward. All the algorithms were tested to be effective and efficient. Themain work is as follows:(1) The recursive mode between topologies is got.Structure properties oftopologies are find. From the perspective of recurrence, topologies of a set having nelements were generated via those of a set having n1elements. Structureproperties of topologies are proved by theorems. So the recursive relationshipbetween topologies became clear.(2)Recursive algorithms were designed to generate topologies on a finite set.Starting with the enumeration, the process to design algorithms was gradual andsequential.Firstly, the enumeration was achieved by recurrence, topologies weregenerated by topology-base and extended-subclusters, composed of extended subset.And the closed nature was used to get the recursive algorithm based onextended-subclusters. Secondly, the concept of quasi-topology was put forward andquasi-topologies were used to construct the closed-clusters, the recursive algorithmbased on quasi-topology was obtained. Finally, with the detailed recurrence andcrystallized recursive procedures, main-subset was extracted as a key structure oftopology, getting the recursive algorithm based on main-subset.(3)The effectiveness of the algorithms were tested by experiments.Experimentsshowed that when the number of elements was eight, it was impossible to generatetopologies by enumeration.But using the algorithm put forward in the article, topologies on a finite set containing eight elements were got within valid time.(4)The efficiency of the algorithms were improved by a great degree. Startingfrom enumeration,three algorithms were put forward to generate topologies. Inreducing the time complexity, significant results were achieved and the efficiency ofthe algorithms were improved by a great degree. |
