Number Of Distinct Topologies On Finite Set With R-Open Sets

Abstract

In this paper, we have studied the sequences of distinct topologies on n-set with r-open sets [2]. Here we have applied the concept of repeated ratios proposed by Dasre and Gujarathi

[3] to these sequences. After applying the repeated ratios to these sequences, we observed that the repeated ratios follow the same pattern for all sequences. So the method of repeated ratios can be used for finding the better bounds for the number of topologies. Using the repeated ratios, one can approximate the large numbers for the sequences as in [3, 4]. We also observed that the sequence of repeated ratios oscillates around one.