Abstract
A topological space X is said to be splittable over a class P of spaces if for every A⊂X there exists continuous f:X→Y∈P such that f(A)∩f(X∖A) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets. We identify precisely which subsets of a poset can be split along over an n-point chain. Using these results it is shown that the union of two splittability classes need not be a splittability class and a necessary condition for P to be a splittability class is given.
Similar content being viewed by others
References
Arhangel'ski?, A. V. (1985) A general concept of cleavability of topological spaces over a class of spaces, in Abstracts Tiraspol Symposium (Stiinca, Kishinev), pp. 8-10 (in Russian).
Arhangel'ski?, A. V. (1993) A survey of cleavability, Topology Appl. 54, 141-163.
Marron, D. J. (1997) Splittability in ordered sets and in ordered spaces, Ph.D. Thesis, Queen's University of Belfast.
Marron, D. J. andMcMaster, T. B. M., Cleavability in semigroups, Semigroup Forum, to appear.
Marron, D. J. and McMaster, T. B. M., Splittability for ordered topological spaces, Boll. Un. Mat. Ital., to appear.
McCartan, S. D. (1979) Minimal T ES-spaces and minimal T EF-spaces, Proc. Roy. Irish Acad. Sect. A 79(2), 11-13.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hanna, A.J., McMaster, T.B.M. Splittability for Partially Ordered Sets. Order 17, 343–351 (2000). https://doi.org/10.1023/A:1006483609936
Issue Date:
DOI: https://doi.org/10.1023/A:1006483609936