Difference of sequence topologies

Pengyu Liu, Jingzhou Na

Published 2022-04-04Version 1

We argue that topology can be interpreted as an area of mathematics studying preserved properties under an equivalence relation, and representation, classification and comparison of the corresponding equivalence classes. With this understanding, we can generalize ideas in topology to non-geometric objects. In this paper, which presents an example of such generalization, we define a sequence topology to be an equivalence class of finite integer sequences of the same length under relabeling or permutations. The difference for a set of finite integer sequences of the same length is defined to be the number of mismatches in the sequences. While the difference for a set of sequence topologies is defined to be the minimum difference over all sets of sequences constructed by choosing one sequence from each sequence topology. We count the number of different sequence topologies of a given length and a set of possible labels and determine the minimum upper bound of the difference for sequence topologies. Finally, we compute the exact difference for a set of sequence topologies of the same length.