Commonalities of Kolmogorov Topologic Spaces and Addition Chains

A table of values m(n) with n ≤ 480 follows:
n1234567891011121314151617181920212223242526272829303132
+001223343445455545565666566667675
+3266767776777777867777878788878886
+6477878887888888978888889898989997
+96888898989998999899999998999999107
+12888989998999999108999999109991091010108
+160999999109109109101010910101091010109101010101010118
+192999910910910101091010109101010101010109101010101010119
+22410101010101010101010111011101191010101010101110101011101111108
+2569910910101091010101010101191010101010101110101011101111119
+288101010101010111010101110111111101110111011111110111111111111119
+3201010101010101110111011101111111011111110111111101111111111111110
+352111111111111111011111111111111101111111111111111111112111111119
+3841010101011101110111111101111111011111111111111101111111111111110
+4161111111111111111111112111111121011111111111112111111121112121110
+448111111111111111111111111121112111211121112121211121212111212 10
Table entries with orange background color mean, that the corresponding ℓ(n) is 1 larger. This data can be obtained by computing all T(k,n) for k ≤ 11. This was done the first time by M. Erné and K. Stege 1990. The single entries 12 can be deduced by the above mentioned two inequalities m(n+2), m(n+1) ≤ m(n)+1, except n=471=3*157 and n=475=5*5*19 for which values the product inequality was used.

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,....


Achim Flammenkamp
Last update: 2020-05-27 13:23:03 UTC+1