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A pointfree study of bases for spaces of minimal prime ideals
Abstract
Let C(X) denote the ring of all real-valued continuous
functions on a topological space X, and mX its space of minimal
prime ideals with the hull-kernel topology. It has a base
consisting of clopen sets and which contains the closure of the union of any
sequence of its members. In the paper Spaces with a pretty base (J. Pure
and Appl. Algebra 70 (1001), 81-87), spaces with a base with these properties
are studied and are shown to have almost all
of the known properties of the space ??
including the fact that if it is a weakly Lindelöf space, then it is basically
disconnected. In the present paper, analogous results are derived in a
pointfree context in which topological spaces are replaced by frames. In some
cases, we are able to obtain more general results with simpler proofs.
Mathematics Subject Classification (2000): Primary: 06D22, 06F25; Secondary: 13A99, 54HXX
Key words: Pretty base, minimal prime ideal, hull-kernel topology, quotient frame, nucleus, pseudocomplement, separated,
weakly Lindelöf, basically disconnected, P-frame cocompact, quasi-complemented.
Quaestiones Mathematicae 26(2003), 333–346
functions on a topological space X, and mX its space of minimal
prime ideals with the hull-kernel topology. It has a base
consisting of clopen sets and which contains the closure of the union of any
sequence of its members. In the paper Spaces with a pretty base (J. Pure
and Appl. Algebra 70 (1001), 81-87), spaces with a base with these properties
are studied and are shown to have almost all
of the known properties of the space ??
including the fact that if it is a weakly Lindelöf space, then it is basically
disconnected. In the present paper, analogous results are derived in a
pointfree context in which topological spaces are replaced by frames. In some
cases, we are able to obtain more general results with simpler proofs.
Mathematics Subject Classification (2000): Primary: 06D22, 06F25; Secondary: 13A99, 54HXX
Key words: Pretty base, minimal prime ideal, hull-kernel topology, quotient frame, nucleus, pseudocomplement, separated,
weakly Lindelöf, basically disconnected, P-frame cocompact, quasi-complemented.
Quaestiones Mathematicae 26(2003), 333–346
