There is an intrigued similarity between the number of finite topologic spaces and the length of shortest addition chains.
Thus let us first explain what a finite topologic space is.

You start with a
finite set *B* of cardinality *k*. Next you consider
its power set *P(B)* which is by definition the set of all the *2 ^{k}*-many possible subsets of

You may translate this definition of finite topologies into binary 0-1-strings of length

Then one may ask:

This unique sequence of numbers

Because there is a on-to-one mapping of finite topologic spaces to preorders of finite sets, we also can consider preorders. Define for any

Notice, because we are interested only in the minimal number of points of a topologic space, we can restrict to non-isomorphic

If you delete both top points the 6 remaining points have no relations and generate

K. Ragnarsson and B. E. Tenner reported in their paper OBTAINABLE SIZES OF TOPOLOGIES ON FINITE SETS from 2008, that M. Erné and K. Stege in their report entitled "Counting finite posets and topologies", Tech. Report 236 , University of Hannover, 1990, already had computed the values *T(k,n)* for * k ≤ 11* and all *n*, which enabled them to answer my question for *n ≤ 379*.
I recomputed these numbers *m(n)* for * n ≤ 379* and can now state that they equal *ℓ(n)* with the exception of 12 values, namely *n ∈ {71, 139, 141, 142, 263, 267, 269, 275, 277, 278, 282, 284}* for which *ℓ(n)=m(n)+1* -- i.e. the minimal number of points *m(n)* needed for a topology of *n* open sets is 1 less than the length of a shortest addition chain for those numbers *n*.

It is tempting to conjecture that *m(n)* is a lower bound for *ℓ(n)*. Sadly computing *m(n)* seems to be much harder than to compute *ℓ(n)*. Thus it seems more useful to conjecture *ℓ(n)* as an upper bound for *m(n)*.

Finally here is a table of *m(n)* with *n ≤ 480* and a list of relations between finite topologic spaces resp. finite posets and addition chains to get those *m(n)* with * 379 ≤ n ≤ 478* from the *T(k,n)* values for * k ≤ 11*.

Last Sunday in October 2016, I was able to compute all *T(k,n)* values for * k ≤ 14* and thus could extend the *m(n)*-sequence up to its 2790th term.

Until 21th November 2016, I was able to compute all *T(15,n)* values, too, and therefore could extend this *m(n)*-sequence up to its 4966th term.

Footnote: *T(k,n)* denotes the number of non-isomorphic *T _{0}*-topologic spaces made up by

Achim Flammenkamp

Last update: 2020-05-27 13:20:09 UTC+2