Spaces N ∪ R and their dimensions

Terasawa, Jun

doi:10.1016/0166-8641(80)90020-6

Cited by 31 publications

(9 citation statements)

References 2 publications

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“…One should not expect that if /3X is an .ES.ff-coinpactin'cation, then Lk{Z(X)) = L(Z(X)). Indeed; by [16…”

Section: N=lmentioning

confidence: 96%

“…The following example shows that the lattice L(Z a (X)) cannot be replaced by L k (Z a (X)) in Theorem 2.7. EXAMPLE 2.8: For a maximal almost disjoint family 1Z of subsets of the set N of positive integers, let NL)7£ denote the set-theoretic union of N and 1Z equipped with the following well-known topology: the points of N are isolated, while a neighbourhood base for a point X £ 71 is the collection {{A}U(A \ F) : F is a finite subset of N} (see [10,51] and [16]). Suppose that V : that there exists a natural isomorphism between CO(<j>X \ X) and £{X); however, their route to this isomorphism seems somewhat obscure for the form of the natural isomorphism was not described explicitly.…”

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confidence: 99%

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Compactifications and quotient lattices of Wallman bases

Wajch

1996

Bull. Austral. Math. Soc.

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We improve the intuition and some ideas of G. D. Faulkner and M. C. Vipera presented in the article "Remainders of compactifications and their relation to a quotient lattice of the topology" [Proc. Amer. Math. Soc. 122 (1994)] in connection with the question about internal conditions for locally compact spaces X and Y under which /3X \ X = f)Y \ Y or, more generally, under which the remainders of compactifications of X belong to the collection of the remainders of compactifications of Y. We point out the reason why the quotient lattice of the topology considered by Faulkner and Vipera cannot lead to a satisfactory answer to the above question. We replace their lattice by the qoutient lattice of a new equivalence relation on a Wallman base in order to describe a method of constructing a Wallman-type compactification which allows us to deduce more complete solutions to the problems investigated by Faulkner and Vipera. INTRODUCTIONAll topological spaces considered below will be locally compact and HausdorfF. The word "space" will be used to refer to such, a topological space, unless otherwise specified.For a space X, the symbol K.(X) will denote the lattice of all compactifications of X, while TZ(X) will stand for the collection of all remainders of X, that is, TZ(X) -{aX \ X : aX £ )C(X)} if we do not distinguish between homeomorphic spaces.Although it is generally known that non-homeomorphic spaces can have homeomorphic remainders of their Cech-Stone compactifications, the nature of such spaces is not well understood. For instance, how to compare such a trivial space as the space N of positive integers with a pseudocompact non-normal space A = /3E \ (/?N \ N) (see [10, 6P])? But the reason for which /3N\ N = /3A \ A must surely be hidden somewhere in the internal structures of the topologies of N and A.The question of when a space K can be homeomorphic to /3X \ X is also nontrivial. Let us mention that, for instance, the statement "every Parovicenko space of weight 2 W is homeomorphic to /3N\N" is equivalent to the continuum hypothesis (see [8]). Furthermore, by Magill's theorem (see [6, 7.2]), the inclusion TZ(X) C ~R{Y)

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“…One should not expect that if /3X is an .ES.ff-coinpactin'cation, then Lk{Z(X)) = L(Z(X)). Indeed; by [16…”

Section: N=lmentioning

confidence: 96%

mentioning

confidence: 99%

Compactifications and quotient lattices of Wallman bases

Wajch

1996

Bull. Austral. Math. Soc.

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“…Let N denote the positive integers and R a maximal family of almost disjoint infinite subsets of N such that fl(N U ~) -(N t.J 7~) = C, where C is the Cantor set. (See [9]; the points of N are isolated and any R E 7~ has neighborhood base {{R} tJ (R -F) IF _C R is finite }.) Let c~(N U ~) indicate the compactification of N U T~ obtained by identifying 0, 1 E C to a point p under a mapping t. Let D be a dense, countable subset of C for which 0, 1 ~ D. Set ~= a(N U T~) -D, C1 = t(C) and ~X = a(N tJ 7~).…”

Section: Proof (A) Implies (B) Let P E R(x) and Let Cp Be The Compomentioning

confidence: 99%

Spaces betweenX and its Freudenthal compactification

Hatzenbuhler

Mattson

1995

Period Math Hung

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Let CX indicate the Freudenthal compactification of a rimcompact, completely regular Hausdorff space X. In this paper the spaces Y which satisfy X C Y C CX are characterized. From this a characterization of when X lies between its locally compact part L(X) and r follows. Such spaces necessarily possess a compactification aX for which Cla x (~X -X) is 0-dimensional. Conditions, including those internal to X, are provided which are necessary and sufficient for this property to hold.

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“…Take any choice of a maximal almost disjoint family on N yielding a Ψ which is almost compact (see [8, 6J]). Such MAD families exist by [17] and [22]. Now βΨ is scattered of CB-index 3 so βΨ is RG and rg(βΨ) ≤ 7.…”

Section: Proof (I) and (Iimentioning

confidence: 99%

On RG-spaces and the regularity degree

Raphael¹,

Woods

2006

Appl.Gen.Topol.

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Abstract. We continue the study of a lattice-ordered ring G(X), associated with the ring C(X). Following [10], X is called RG when G(X) = C(X δ ). An RG-space must have a dense set of very weak P-points. It must have a dense set of almost-P-points if X δ is Lindelöf, or if the continuum hypothesis holds and C(X) has small cardinality. Spaces which are RG must have finite Krull dimension when taken with respect to the prime z-ideals of C(X). There is a notion of regularity degree defined via the functions in G(X). Pseudocompact spaces and metric spaces of finite regularity degree are characterized.2000 AMS Classification: Primary 54G10; Secondary 46E25, 16E50.

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Spaces N ∪ R and their dimensions

Cited by 31 publications

References 2 publications

Compactifications and quotient lattices of Wallman bases

Compactifications and quotient lattices of Wallman bases

Spaces betweenX and its Freudenthal compactification

On RG-spaces and the regularity degree

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