Stellenbosch University
Compactifications and separation of the interval
topology
Cerene Rathilal\(^*\), University of
Johannesburg
Andrew Craig, University of
Johannesburg
SAMS Subject Classification Number: 13, 34.
In 1942, Frink [2] showed that a lattice is complete iff the interval topology of the lattice is compact. We aim to discuss conditions under which the Dedekind–MacNeille completion of a lattice \(L\) is a compactification of the interval topology on \(L\). In [1], Ernè established conditions for which the interval topology of \(L\) will be Hausdorff and Regular. The concept of finite separabilty plays a central role in understanding the separation axioms of the interval topology of \(L\). As a consequence of investigating compactifications of the interval topology of \(L\), we shall discuss the relationship between separated intervals of a lattice as described in [3], finite separabilty and Ernè’s separation conditions from [2].
References
[1] M Erné, Separation axioms for interval topologies, Proceedings of the Amer. Math Soc., 79 (1980), 185–190.
[2] O Frink, Topology in lattices, Trans. Amer. Math. Soc. 51 (1942), 569–582.
[3] Z Riečanová, Lattices and quantum logics with separated intervals, atomicity, Internat. J. Theoret. Phys. 37 (1998), 191–197.
