Interval Topology Research Articles

A Note On The Separability Of An Ordered Space

An open interval of a simply ordered set S is a subset I of S such that either(1) for some ,(2) for some , or(3) for some .A simply ordered set with its interval topology (i.e., the topology in which “neighborhood of x” means “open interval containing x”) will be called an ordered space.

The Interval Topology of a Lattice

The Interval Topology of a Lattice

The interval topology of a lattice

The interval topology of a lattice

A Boolean algebra without proper automorphisms

It is the purpose of this note to show that there exists an infinite Boolean algebra which has no proper automorphisms.' We shall construct a simply ordered set S, introduce a topology on this set in the usual manner, the so-called interval topology determined by the ordering relation,2 and prove that S is a compact zero-dimensional Hausdorff space and that the only homeomorphism on S onto S is the identity mapping. It is well known3 that the group of automorphisms of the set-field 3 consisting of all open and closed subsets of S is isomorphic to the group of all homeomorphisms on S onto S, whence it follows that B3 has no proper automorphisms. Consider a simply ordered set S with at least two elements. By the interval topology on S we mean the topology which has as a subbasis for open sets, the family of all sets UCS such that either