Details
| Original language | German |
|---|---|
| Pages (from-to) | 221-259 |
| Number of pages | 39 |
| Journal | Manuscripta mathematica |
| Volume | 11 |
| Issue number | 3 |
| Publication status | Published - Sept 1974 |
Abstract
Every topology τ on the finite set X={x1,...,xN} has a minimal base Μ(τ); separation axioms and connectivity are simply characterized; further we obtain a recursive construction of finite topologies. In the second section we give some formulae connecting A(N) (the number of topologies on X), Z(N) (the number of connected topologies on X) and associated numbers. The theory of representation matrices discussed in the third section leads to an easy description of many topological notions as closure operator, induced topology, connectivity; it also yields some more combinatorial formulae. In the last part we improve the remainder term R(N)=Id A(N)-1/4(N)2=0(N3/2ld N) given by Kleitman and Rothschild to 0(N·ld2N). Then, among many other asymptotical results, we show that A(N) and Z(N) are asymptotically equal.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Manuscripta mathematica, Vol. 11, No. 3, 09.1974, p. 221-259.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Struktur- und anzahlformeln für topologien auf endlichen mengen
AU - Erné, Marcel
PY - 1974/9
Y1 - 1974/9
N2 - Every topology τ on the finite set X={x1,...,xN} has a minimal base Μ(τ); separation axioms and connectivity are simply characterized; further we obtain a recursive construction of finite topologies. In the second section we give some formulae connecting A(N) (the number of topologies on X), Z(N) (the number of connected topologies on X) and associated numbers. The theory of representation matrices discussed in the third section leads to an easy description of many topological notions as closure operator, induced topology, connectivity; it also yields some more combinatorial formulae. In the last part we improve the remainder term R(N)=Id A(N)-1/4(N)2=0(N3/2ld N) given by Kleitman and Rothschild to 0(N·ld2N). Then, among many other asymptotical results, we show that A(N) and Z(N) are asymptotically equal.
AB - Every topology τ on the finite set X={x1,...,xN} has a minimal base Μ(τ); separation axioms and connectivity are simply characterized; further we obtain a recursive construction of finite topologies. In the second section we give some formulae connecting A(N) (the number of topologies on X), Z(N) (the number of connected topologies on X) and associated numbers. The theory of representation matrices discussed in the third section leads to an easy description of many topological notions as closure operator, induced topology, connectivity; it also yields some more combinatorial formulae. In the last part we improve the remainder term R(N)=Id A(N)-1/4(N)2=0(N3/2ld N) given by Kleitman and Rothschild to 0(N·ld2N). Then, among many other asymptotical results, we show that A(N) and Z(N) are asymptotically equal.
UR - http://www.scopus.com/inward/record.url?scp=19544362788&partnerID=8YFLogxK
U2 - 10.1007/BF01173716
DO - 10.1007/BF01173716
M3 - Artikel
AN - SCOPUS:19544362788
VL - 11
SP - 221
EP - 259
JO - Manuscripta mathematica
JF - Manuscripta mathematica
SN - 0025-2611
IS - 3
ER -