Lattice-equivalence of topological spaces

Thron, W. J.

doi:10.1215/s0012-7094-62-02968-x

Cited by 57 publications

(34 citation statements)

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“…When (X, #) is also a T 0 -space we say that it is a T 0D -space. In this case the definition reduces to that in Thron [4], since a !T 0 -space (X, #) is a T 0O -space if and only if the derived set {a;}' = %\{x} is closed for each x in X. We prove…”

Section: Theorem (21) In Order That Two T-spaces (3tmentioning

confidence: 99%

On the lattice equivalence of topological spaces

Finch

1966

J. Aust. Math. Soc.

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A topology on a set X is defined by specifying a family of its subsets which has the properties (i) arbitrary set intersections of members of belong to , (ii) finite set unions of members of belong to and (iii) the empty set □ and the set X each belong to . The members of are called the closed subsets of X. If X is any subset of X then denotes the closure of X, that is, the set intersection of all closed subsets which contain X, however when X = {x} contains one point only we will denote by . The pair (X, ) is called a topological space or, in what follows, a T-space. By a T-lattice we mean a complete distributive lattice of sets in which arbitrary g.l.b. means arbitrary set intersection, finite l.u.b. means finite set union and which contains the empty set □ It is well-known, for example Birkhoff [1], that if (X, ) is a T-space and the members of are partially ordered by set inclusion then is a T-lattice.

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Section: Theorem (21) In Order That Two T-spaces (3tmentioning

confidence: 99%

On the lattice equivalence of topological spaces

Finch

1966

J. Aust. Math. Soc.

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“…A lattice (T, S;) will be called a "»f-lattice iff it is complete, distributive, and has a base consisting of irreducible elements. Thron [2] proved that a lattice has a topological representation iff it is a 'tf-lattice. Other characterization had previously been given by other authors.…”

Section: [October Is An Isomorphism From (T S:) Onto (#2)mentioning

confidence: 99%

“…Other characterization had previously been given by other authors. For references to these as well as to other results related to our subject the reader is referred to [2].…”

Section: [October Is An Isomorphism From (T S:) Onto (#2)mentioning

confidence: 99%

“…A space satisfies the 7B-axiom iff for every xeX the set [x]', the derived set of [x], is closed. Using this axiom, Thron [2] proved the theorem that, in our terminology, the subfamily of TD-spaces in a representation family of a 1^-lattice contains at most one element. This result is easily obtained from our discussion once the following theorem has been established.…”

Section: [October Is An Isomorphism From (T S:) Onto (#2)mentioning

confidence: 99%

“…Part (1) follows from the fact that the set of necessary closures gets mapped onto itself by all automorphisms of F. To prove part (2) assume that the lattice T has a least representation space (X*/€*) under 2:**, in a representation family. Assume further that the set 2(fë*) contain an occasional closure C. Then C can be omitted from the base &(*€*) and the remaining point closures form a base 2$' of <€*, by Lemma 3.2.…”

Section: [October Is An Isomorphism From (T S:) Onto (#2)mentioning

confidence: 99%

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