… to partitions of sets or numbers, respectively, is evident: labeled topologies create set partitions
… to homeomorphism, finite topologies are related, in a similar way, to partitions of integers, …

… topology as we have shown this recently in [62n]: There is a left-separated nonmetrizable
space with point-countable base which is perfectly normal and paracompact in any of its finite …

… A noncommutative and noncocommutative Hopf algebra on finite topologies H … -partitions
associated to a special poset, we define the notion of T-partitions associated to a finite topology…

… in which the summaLion is taken over all 2 n-1 ordered partitions … (pz,...,pm) is the number
of T0-topologies of a certain special form, connected with the partition n = Pz +..- + Pm- For V(pl…

… -topologies on n points with the number of topologies on n points; (ii) the number of connected
T0-topologies on … By irn we mean the set of all unordered partitions of n. If к e nn, by к. we …

… It follows that A is dense in every κ-bounded topology on G. Hence, if G can be partitioned …
For a partition P of G, we denote by P∧P−1 the partition of G into the cells {A∩B−1 : A, B ∈ P}…

… A topology on X is called a partition topology if it possesses a base which is a … If X is a finite
set, the lattice of regular topologies on X is precisely the lattice of partition topologies on X. [46…

… To the unique partition with ~=i there corresponds in the formula for ~o(~] the summand ~V(~…
[ , where p is a prime, we have the partition with n~--i and two other partitions (~i] and C[~p) …

… By a natural level order, we mean a partial order on n whose level partition satisfies Li = {WI-,+I,...,
m, } where m, = 0 and mi = mi- , + ni. It is evident that such a partial order is always a …

… of the m poirtts of C one of the par~s of a given partition of V~ into m parts. Thus, there are
S('*~, … those integers which appear in that part of the partition of V,~ which is assigned to the rth …