The Largest Cardinals of Isomorphic Class of Topology-set-lattices on an Infinite Set

Let X be an infinite set,K={τ:τ is a topology on X},defineτ≌σ iff (f)(f is a lattice-isomorphism from τ to σ),for a given τ∈K we define τ={σ:σ∈K ＆σ≌τ}K(K)is an imcomparable class iff (τ∈K)(σ∈K)(τ≠→τ and σ are incomparable),M={K:K is an incomparable class}. Theorem 1 sup{|τ|:τ∈K}=max{|τ|:τ∈k}=|K|=2~2~X]=exp(exp(|X|)). Theorem 2 sup{sup{|τ|:τ∈K}:KK ＆ K is an incomparable class}=sup