Commonalities of Kolmogorov Topologic Spaces and Addition Chains

• forall n ∈ N:
  ℓ(n+1) ≤ ℓ(n)+1
  m(n+1) ≤ m(n)+1
• forall n ∈ N \ {1}:
  ℓ(n+2) ≤ ℓ(n)+1
  m(n+2) ≤ m(n)+1
• forall n,h ∈ N:
  ℓ(n*h) ≤ ℓ(n)+ℓ(h)
  m(n*h) ≤ m(n)+m(h)
• forall n ∈ N with ν(n) ∈ {1,2} :
  there is a Kolmogorov topologic space with n open sets and at least k points
  there is a shortest addition chain to n of length k.
• forall n ∈ N:
  ℓ(n) ≤ λ(n)+ν(n)-1
  m(n) ≤ λ(n)+ν(n)-1

Similar to the function c(k) for addition chains of length k there is the function f(k) for finite topologies defined: to be the smallest number of open sets which needs at least k points. This sequence starts like 1,2,3,5,7,11,19,29,47,79,127,191,379,... -- see also A137814 in OEIS.
Moreover we can also define the function t(k) similar to d(k) for shortest addition chains, to be the number of sizes of finite topologies which need at least k points. Those sequence looks like 1,1,2,3,5,9,15,26,45,80,141,256,....