On the enumeration of finite maximal connected topologies

Shawpawn Kumar Das
1973 Journal of combinatorial theory. Series B (Print)  
A connected topology F is said to be maximal connected if ?# strictly finer than Y implies that % is disconnected. In this paper, it is shown that the number of homeomorphism classes of maximal connected topologies defined on a set with n points is equal to twice the number of n point trees minus the number of n point trees possessing a symmetry line. An enumeration of a class of topologies, called critical connected topologies, which includes the maximal connected spaces is then carried out
more » ... h the help of Polya's theorem. Another result is that a chain of connected n point T" topologies, linearly ordered by strict fineness, can contain a maximum of $(n" -3n + 4) topologies, and, moreover, this number is the best possible upper bound for the length of such a chain.
doi:10.1016/0095-8956(73)90020-8 fatcat:vnp6ibdcxnecjap4oms2bsdz5m