A Machine Representation of Finite T 0 Topologies
Shawpawn Kumar Das · Journal of the ACM · 1977
ASSrRACr This paper introduces the concept of stmdartty, which defines an equivalence relation on the family of labeled To topologms on a fimte set Two slmdar spaces are necessarily homeomorphic, but the converse Is not true in general It is shown that the ratio of the number of similarity classes of labeled n-point To topologies to the number of labeled To spaces on n points goes asymptotically to zero for large n S~mdanty classes can be described completely by a partmoned rectangular integer array entity named APARNA.The APARNA representation is suitable for manipulation by a digital computer and, as it clearly displays certain structural aspects, appears to be useful in the computer study of finite topologies An APARNA-based computer algorithm for the enumeration of labeled To topologies has generated the numbers of such spaces definable on sets with 8, 9, 10, and 11 points.These numbers are, respectively, 431,723,379; 44,511,042,511; 6,611,065,248,783, and 1,396,281,677,105,899 The corresponding numbers for all possible labeled topologies (To as well as non-T0) are, respectively, 642,779,354, 63,260,289,423, 8,977,053,873,043; and 1, 816,846,038,736,192