Linearly ordered extensions of GO spaces

Miwa, Takuo; Kemoto, Nobuyuki

doi:10.1016/0166-8641(93)90057-k

Cited by 20 publications

(12 citation statements)

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“…(3) SinceM has a dense subspace M × {0} (homeomorphic to M) which is not perfect,M is not perfect and thus it is not metrizable. By [5]C is not perfect and thus not metrizable. Since M and C have the underlying LOTS R, by Theorem 2.1, the projection mappings φ 1 :M → R and φ 2 :C → R are (finite-to-one) closed mappings.…”

Section: Resultsmentioning

confidence: 99%

“…It is well known that paracompactness, metrizability and quasi-developability of X are preserved by X * . However, these properties of X are not hereditary toX (see [5]). Since the Sorgenfrey line S and the Michael line M are submetrizable they have the continuous Urysohn property P. M * has P [2], but S * andS do not have P [8].…”

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confidence: 99%

“…By [5,Proposition 2.7],X is the minimal dense linearly ordered extension of X and, by [7,Theorem 9], X * is the minimal closed linearly ordered extension of X .…”

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confidence: 99%

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A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

Gao

Shi

2011

Bull. Aust. Math. Soc.

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For any generalized ordered space X with the underlying linearly ordered topological space X u , let X * be the minimal closed linearly ordered extension of X andX be the minimal dense linearly ordered extension of X . The following results are obtained.(1)The projection mapping π :The projection mapping φ :X → X u , φ( x, i ) = x, is closed. 2010 Mathematics subject classification: primary 54C10; secondary 54C25, 54D20, 54D30, 54F05.

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Section: Resultsmentioning

confidence: 99%

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confidence: 99%

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A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

Gao

Shi

2011

Bull. Aust. Math. Soc.

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“…One of these is X * and was defined by D. J. Lutzer [7]. The other one is L(X) and was studied in [8]. We review here the definitions of those linearly ordered extensions.…”

Section: Two Linearly Ordered Extensions and Notationmentioning

confidence: 99%

“…It is easily seen that X * contains X as a closed subset and L(X) contains X as a dense subset. See [7,8] for further information about X * and L(X). In both cases, X and X × {0} are identified by the correspondence of x to (x, 0).…”

Section: S δ -Diagonals In Linearly Ordered Extensionsmentioning

confidence: 99%

Dense Sδ-diagonals and linearly ordered extensions

Hosobuchi

2003

Appl. Gen. Topol.

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Abstract. The notion of the S δ -diagonal was introduced by H. R.Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an S δ -diagonal and we showed that the S δ -diagonal property is stronger than the quasi-G δ -diagonal property. In this paper, we define a dense S δ -diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense S δ -diagonals if the sets of right and left looking points are countable.Keywords: S δ -diagonal, dense S δ -diagonal, linearly ordered space (LOTS), generalized ordered space (GO-space), linearly ordered extension.2000 AMS Classification: 54F05. S δ -diagonalsWe review in this section the definitions of S δ -set and S δ -diagonal, and state our results obtained in [5].The following definition is a generalization of a G δ -set and was introduced by H. R. Bennett [2] to study the quasi-developability of linearly ordered (topological) spaces. Definition 1.1. Let X be a topological space. A subset A of X is called an S δ -set if there exists a countable collection {U (1), U (2), . . .} of open subsets of X such that, for two points p ∈ A and q ∈ X \ A, there exists an n such that p ∈ U (n) and q / ∈ U (n).It is easy to see that a G δ -set is an S δ -set. Hence the notion of S δ -set is a generalization of G δ -set. See [3] for a description of S-normal spaces whose closed subsets are S δ -sets.

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