A note on compactifications and semi-normal spaces

Alò, Richard A.; Shapiro, Harvey L.

doi:10.1017/s1446788700004638

Cited by 17 publications

(7 citation statements)

References 9 publications

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“…Since every Wallman-Frink compactification is an Alexandroff base compactification (corollary to Theorem 6), it follows that the Stone-Cech compactification is always obtainable in this way and, for locally compact Hausdorff spaces, the one-point compactification is an Alexandroff base compactification. More generally, we can appeal to the results of Njastad [12] and Alo and Shapiro [3] to conclude that the compactifications of Freudenthal [10], Fan-Gottesman [6] and Gould are all Alexandroff-base compactifications.…”

Section: Discussionmentioning

confidence: 99%

Compactifications

Wasileski¹

1974

Can. j. math.

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Every completely regular space has at least one Hausdorff compactification and much research in Topology has been devoted to methods of constructing the compactifications of completely regular spaces. These methods fall into two general categories: internal methods and external methods. External methods are characterized by their reliance on structures which are not topological or outside the immediate topological structure of the space in question; examples of the former are A. Weil [20] - uniform structures and Yu. Smirnov [15] who uses the proximity structures of Efremovich; using the continuous real-valued functions to produce an embedding of the space serves as an example of the latter.

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Section: Discussionmentioning

confidence: 99%

Compactifications

Wasileski¹

1974

Can. j. math.

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“…Alo and Shapiro [2]* used another approach for the results. While Alo and Shapiro in [1]* imposed some conditions on the normal base ^ (see Theorem 2, [1]), and gave similar results for a wider class of compactifications, Njastad showed that Alexandroff, StoneCech, Freudenthal [6], Fan-Gottesman [5], and Gould [9] compact'-fications satisfy the conditions in his theorem.…”

mentioning

confidence: 89%

“…(1) We know that there exists a space E such that I is incompletely regular. For example, let E ι be any Hausdorff space.…”

Section: If E a Hausdorjf Space Is Such That I = [0 1] With The Usmentioning

confidence: 99%

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Wallman compactifications onE-completely regular spaces

Piacun¹,

Su²

1973

Pacific J. Math.

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“…Before stating our main results we will give three lemmas that will be needed. The proof of Lemma 1 can be found in [1]. LEMMA…”

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confidence: 99%

ℒ-Realcompactifications and Normal Bases

Alò

Shapiro

1969

J. Aust. Math. Soc.

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In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular 7\ spaces X. His method utilized the notion of a normal base. A normal base 2£ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of 2£'.Frink showed that if X has a normal base, then the Wallman space, (a{21£), consisting of the i^-ultrafilters is a Hausdorff compactification of X. This also showed that X must be a Tychonoff space. In this note we use the notion of ^-ultrafilters in a countably productive normal base 2£ to introduce a new space rj(^) consisting of all those iF-ultrafilters with the countable intersection property.Every normal base 3£ of X corresponds to a normal base 2£* in r]{2£) (and also in co(2?)). We show that every collection of J?*-ultrafilters with the countable intersection property is fixed, that is the intersection of all the members of the collection is non empty. In light of this fact, we say that r\{2?) is 3£*-real-compact. We also show that r)(^) is contained in thê -closure of X in ay(2£). Finally if 2£ is the collection of all zero-sets then r\{2£) is precisely the Hewitt real compactification of X. We have attempted to show that every realcompactification Y of a space X can be obtained as a space rj(^). This remains an open question.Many examples exist of normal bases which are countably productive. One of the most important is the collection of all zero-sets of a completely regular T 1 space. Gillman and Jerison in [3] have shown that this family is countably productive and also that it satisfies the requirements for a normal base. Thus every Tychonoff space has a countably productive normal base. DEFINITIONS. A base $£ for the closed sets of a T 1 space X is said to be disjunctive if given any closed set F and any point x not in F there is a closed set A of 2£ that contains x and is disjoint from F. The base is said to be normal if any two disjoint members A and B of 3£ are subsets respectively of disjoint complements C" and D' of members of 2£.A family 2£ of subsets of a set X is a ring of sets if it is closed under 489 use, available at https://www.cambridge.org/core/terms. https://doi

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A note on compactifications and semi-normal spaces

Cited by 17 publications

References 9 publications

Compactifications

Compactifications

Wallman compactifications onE-completely regular spaces

ℒ-Realcompactifications and Normal Bases

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