Vol. 28, No. 2, 1969

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Hypotopological spaces and their embeddings in lattices with Birkhoff interval topology

Kirby Alan Baker

Vol. 28 (1969), No. 2, 275–288

Important classes of topological spaces have topologies which are induced by a generating collection of closed subsets; typical examples are k-spaces, sequential spaces with unique sequential limits, and lattices with the Birkhoff interval topology. This paper proceeds by axiomatizing this construction—a set with a specified generating collection of closed subsets is called a “hypotopological space.” The Birkhoff interval topology is then studied in these terms. A natural embedding of hypotopological spaces in conditionally complete, atomic, distributive lattices with Birkhoff interval topology is derived. This embedding is used to show that lattices with Birkhoff interval topology have the same nontrivial subspace and product properties as k-spaces and sequential spaces. In particular, we answer in the negative a question first raised by Birkhoff, namely, whether the Birkhoff interval topology is preserved under the formation of the product of two lattices.

Mathematical Subject Classification
Primary: 54.10
Secondary: 06.00
Received: 17 January 1968
Published: 1 February 1969
Kirby Alan Baker