Abstract
In this article, $ Top(X_{n}) $ is the collection of all topologies on a given set $X_{n}$ having cardinality $n$. We introduce the method to find the number of topologies of finite sets of cardinality up to $5$ by combinations and counting principles, which enable us to enumerate the classes of homeomorphic disconnected topologies as well as the connected topologies. Moreover we study the graphical aspects of the connected and disconnected topologies.
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