Separation Axioms Between T0 and T1

Aull, C. E.; Thron, W. J.

doi:10.1016/s1385-7258(62)50003-6

Cited by 89 publications

(75 citation statements)

References 3 publications

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“…Let t/be a set carrying a topology T, let B = P{U) with the obvious boolean algebra on B, and let d be the derivative operation of T. Then d satisfies (0, 1) but not necessarily (2)…”

Section: Lemma 4 Each Pseudo Topological Space Carries An Associatementioning

confidence: 99%

Topological aspects of suitable theories

Simmons

1975

Proceedings of the Edinburgh Mathematical Society

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Roughly speaking a suitable theory is a theory T together with its formal provability predicate Prv (.). A pseudo-topological space is a boolean algebra B which carries a derivative operation d and its associated closure operation c. Thus we can pretend that B is a topological space. We show that the Lindenbaum algebra B(T) of a suitable theory becomes, in a natural way, a pseudotopological space, and hence we can translate properties of T into topological language, as properties of B(T). We do this translation for several properties of T, including (1) satisfying Gödel's first theorem, (2) satisfying Löb's theorem and (3) asserting one's own inconsistency. These correspond to the topological properties (1) having an isolated point, (2) being scattered, (3) being discrete.

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“…Let t/be a set carrying a topology T, let B = P{U) with the obvious boolean algebra on B, and let d be the derivative operation of T. Then d satisfies (0, 1) but not necessarily (2)…”

Section: Lemma 4 Each Pseudo Topological Space Carries An Associatementioning

confidence: 99%

Topological aspects of suitable theories

Simmons

1975

Proceedings of the Edinburgh Mathematical Society

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“…Example 4 is an example of a minimal T o topology on an infinite set which is bicompact, and a similar example can be given for minimal TVspaces. Note that in Example 4, [1] is a closed set. This leads to the following theorem.…”

Section: // (X J/~) Is a Topological Space In Which The Open Sets Armentioning

confidence: 99%

MinimalT₀-spaces and minimalT_D-spaces

Larson¹

1969

Pacific J. Math.

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“…Since the necessary closures form a base of F2n) there are exactly 2" distinct bases of irreducible elements in r(2") and hence m(F2n)) = 2". Aull and Thron [1] recently introduced a new separation axiom, called the Tß-axiom, which is strictly stronger than T0 and strictly weaker than T,. A space satisfies the 7B-axiom iff for every xeX the set [x]', the derived set of [x], is closed.…”

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

confidence: 99%