Characterizations Of The Interval Topology On Posets

In this paper , we investigate the interval topology on a poset . We show that a lattice homomorphism between two complete lattices is continuous with respect to the interval topologies if and only if the lattice homomorphism preserves arbitrary nonempty infs and sups .It is easily verified that the interval topologyθ(L) of a direct product L of posets L_j is always contained in the cartesian product ofθ(L_j). But the conversely inclusion is not always correct . In this paper we show that the interval topology of a direct product of bounded posets is equivalent to the cartesian topology .In chapter 4, we give some characterizations of compactness of interval topology . The last part of this paper gives some characterizations of relationships between the lattices of topologies on a poset.