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Is the Construct of l-Topological Spaces a Co-Tower Extension of Some Simpler Construct?
Abstract
In an earlier paper, the second author associated with any
fibre-small topological construct A and any completely distributive
lattice L, an extension of A, called the (L)-co-tower
extension of A. He demonstrated that many familiar constructs
in fuzzy topology can be expressed as co-tower extensions of more basic constructs
and asked
whether the construct L-Top of L-topological spaces (the stratified
Chang-Goguen spaces) can be expressed as a co-tower extension. In this note
we answer this question with "no" and "yes". In particular we present the following
theorems:
(1) If L is a 3-element linearly ordered lattice (or, more generally,
any linearly ordered finite lattice with at least 3 elements), then L-Top cannot be expressed as an L-co-tower extension.
(2) If L is a 4-element Boolean algebra (or, more generally, any
atomic complete Boolean algebra), then L-Top is (up to concrete
isomorphism) the L-co-tower extension of the construct Top of
topological spaces.
Mathematics Subject Classification (1991): 54B30, 54A40, 18B30
Keywords: topological construct, L-Topological space, co-tower extension,
completely distributive lattice, categorical methods, fuzzy topology, categories
of topological spaces and continuous mappings, distributive, lattice, extensions,
Boolean algebra, algebra, complete, topological space
Quaestiones Mathematicae
24(2) 2001, 147-155
fibre-small topological construct A and any completely distributive
lattice L, an extension of A, called the (L)-co-tower
extension of A. He demonstrated that many familiar constructs
in fuzzy topology can be expressed as co-tower extensions of more basic constructs
and asked
whether the construct L-Top of L-topological spaces (the stratified
Chang-Goguen spaces) can be expressed as a co-tower extension. In this note
we answer this question with "no" and "yes". In particular we present the following
theorems:
(1) If L is a 3-element linearly ordered lattice (or, more generally,
any linearly ordered finite lattice with at least 3 elements), then L-Top cannot be expressed as an L-co-tower extension.
(2) If L is a 4-element Boolean algebra (or, more generally, any
atomic complete Boolean algebra), then L-Top is (up to concrete
isomorphism) the L-co-tower extension of the construct Top of
topological spaces.
Mathematics Subject Classification (1991): 54B30, 54A40, 18B30
Keywords: topological construct, L-Topological space, co-tower extension,
completely distributive lattice, categorical methods, fuzzy topology, categories
of topological spaces and continuous mappings, distributive, lattice, extensions,
Boolean algebra, algebra, complete, topological space
Quaestiones Mathematicae
24(2) 2001, 147-155
