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 SM ISO690:2012CALMUŢCHI, Laurenţiu. Generalized Hausdorff compactifications. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, pp. 138-139. ISBN 978-9975-81-074-6. |
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| Conference on Applied and Industrial Mathematics Ediţia a 29, 2022 |
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A generalized extension or a g-extension of a space X is a pair (Y, f), where f : X → Y is a continuous mapping and the set f(X) is dense in Y . If f is an embedding, then Y is an extension of X and it is assumed that X = f(X) and f(x) = x for any x ∈ X. Let GE(X) be the set of all g-extension of the space X and E(X) be the set of all extensions of X. Obviously, E(X) ⊆ GE(X). If P is a topological property, then PGE(X) is the set of all g-extensions with the property P and PE(X) = E(X) ∩ PGE(X). A set L ⊆ PGE(X) is a complete lattice of g-extensions of X if for every non-empty set H ⊆ L we have (∨H)∩L ̸= ∅ and (∧H)∩L ̸= ∅. Let X be a T0-space. Let us denote by HGC(X) the set of the gcompactifications (bX, bX) of the space X for which bX is a Hausdorff space.Theorem 1. The set of HGC(X) is a complete lattice of g-extensions with the maximal element. The maximal element of the lattice HGC(X) is denoted by (βX, βX) and is called g-compactification Stone-˘Cech. Corollary 1. If f : X → Y is a continuous application, then there is a single continuous application βf : βX → βY for which βf ◦ βX = βY ◦ f. Corollary 2. If f : X → Y is a continuous application of the space X in the Hausdorff space and compact Y , then there is a single continuous application βf : βX → Y for which f = βf ◦ βX. Theorem 2. ([2], for T1-spaces). For any continuous application of space X in a Hausdorff and compact space Y , there is a single continuous application ωf : ωX → Y for which f = ωX|X. |
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<?xml version='1.0' encoding='utf-8'?> <oai_dc:dc xmlns:dc='http://purl.org/dc/elements/1.1/' xmlns:oai_dc='http://www.openarchives.org/OAI/2.0/oai_dc/' xmlns:xsi='http://www.w3.org/2001/XMLSchema-instance' xsi:schemaLocation='http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd'> <dc:creator>Calmuţchi, L.I.</dc:creator> <dc:date>2022</dc:date> <dc:description xml:lang='en'><p>A generalized extension or a g-extension of a space X is a pair (Y, f), where f : X → Y is a continuous mapping and the set f(X) is dense in Y . If f is an embedding, then Y is an extension of X and it is assumed that X = f(X) and f(x) = x for any x ∈ X. Let GE(X) be the set of all g-extension of the space X and E(X) be the set of all extensions of X. Obviously, E(X) ⊆ GE(X). If P is a topological property, then PGE(X) is the set of all g-extensions with the property P and PE(X) = E(X) ∩ PGE(X). A set L ⊆ PGE(X) is a complete lattice of g-extensions of X if for every non-empty set H ⊆ L we have (∨H)∩L ̸= ∅ and (∧H)∩L ̸= ∅. Let X be a T0-space. Let us denote by HGC(X) the set of the gcompactifications (bX, bX) of the space X for which bX is a Hausdorff space.</p><p>Theorem 1. The set of HGC(X) is a complete lattice of g-extensions with the maximal element. The maximal element of the lattice HGC(X) is denoted by (βX, βX) and is called g-compactification Stone-˘Cech. Corollary 1. If f : X → Y is a continuous application, then there is a single continuous application βf : βX → βY for which βf ◦ βX = βY ◦ f. Corollary 2. If f : X → Y is a continuous application of the space X in the Hausdorff space and compact Y , then there is a single continuous application βf : βX → Y for which f = βf ◦ βX. Theorem 2. ([2], for T1-spaces). For any continuous application of space X in a Hausdorff and compact space Y , there is a single continuous application ωf : ωX → Y for which f = ωX|X.</p></dc:description> <dc:source>Conference on Applied and Industrial Mathematics (Ediţia a 29) 138-139</dc:source> <dc:title>Generalized Hausdorff compactifications</dc:title> <dc:type>info:eu-repo/semantics/article</dc:type> </oai_dc:dc>
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