On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups
<p>This work has largely grown out of the papers of R. Haydon (Studia Math. 52 (1974), 23-31) and E.V. Shchepin (Russian Math. Surveys 31 : 5 (197&), 155-191) Among other results affirmative answers to problems 3, 18, 20 and 21 posed by A. Pełczyński in his monograph (Diss. Math. 58, W...
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| Format: | Thesis |
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1978
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| _version_ | 1797090137604096000 |
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| author | Hoffmann, B |
| author_facet | Hoffmann, B |
| author_sort | Hoffmann, B |
| collection | OXFORD |
| description | <p>This work has largely grown out of the papers of R. Haydon (Studia Math. 52 (1974), 23-31) and E.V. Shchepin (Russian Math. Surveys 31 : 5 (197&), 155-191) Among other results affirmative answers to problems 3, 18, 20 and 21 posed by A. Pełczyński in his monograph (Diss. Math. 58, Warszawa 1968) are given.</p> <p>For pairs (m,n) of extended integers (-1≤m≤n≤∞) the surjective and injective notions of (m,n) -soft and (m,n)-hard maps respectively are introduced and investigated. It is shown that for a map X → Y of compact spaces (-1,0)-softness (resp. (0,0)-hardness) entails the existence of a regular averaging (resp. extension) operator and that the converse holds, if X is metrizable (resp. Y is a Dugundji space). (0,0)-hardness, (-1,0)-softness and (0,0)-softness are shown to be local properties and it is deduced that each point of a Dugundji space and of a quotient space of a locally compact group has a neighbourhood base of Dugundji spaces. Moreover, we prove that a space and its cone are Dugundji spaces simultaneously. (∞,∞)-soft group homomorphisms are isomorphisms. This follows from the result that a compact contractible group is trivial.</p> <p>A compact space is classified as an absolute extensor for m- to n-dimensional spaces or AE(m,n), if the constant map is (m,n)~soft.</p> <p>Equivalently, a compact space is an AE(m,n) if and only if every embedding into a compact space is (m,n)-hard. Haydon's paper entails coincidence of the notions of AE(0,0) and Dugundji space. Here an AE(0,0) (resp. AE(0,∞)) is surjectively characterized as (-1,0)-soft image of a generalized Cantor set (resp. Tychonoff cube). An in- jective characterization of the class of Poano spaces is given as metrizable AE(0,n), n (1≤n≤∞) being arbitrary.</p> <p>Using combinatorial techniques we obtain some results concerning the countable chain condition on spaces of probability measures and spaces of ⩽ n-element subsets.</p> |
| first_indexed | 2024-03-07T03:14:12Z |
| format | Thesis |
| id | oxford-uuid:b5393a9e-8c18-4c66-bffb-6000b54dd37e |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:14:12Z |
| publishDate | 1978 |
| record_format | dspace |
| spelling | oxford-uuid:b5393a9e-8c18-4c66-bffb-6000b54dd37e2022-03-27T04:31:52ZOn a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:b5393a9e-8c18-4c66-bffb-6000b54dd37ePolonsky Theses Digitisation Project1978Hoffmann, B<p>This work has largely grown out of the papers of R. Haydon (Studia Math. 52 (1974), 23-31) and E.V. Shchepin (Russian Math. Surveys 31 : 5 (197&), 155-191) Among other results affirmative answers to problems 3, 18, 20 and 21 posed by A. Pełczyński in his monograph (Diss. Math. 58, Warszawa 1968) are given.</p> <p>For pairs (m,n) of extended integers (-1≤m≤n≤∞) the surjective and injective notions of (m,n) -soft and (m,n)-hard maps respectively are introduced and investigated. It is shown that for a map X → Y of compact spaces (-1,0)-softness (resp. (0,0)-hardness) entails the existence of a regular averaging (resp. extension) operator and that the converse holds, if X is metrizable (resp. Y is a Dugundji space). (0,0)-hardness, (-1,0)-softness and (0,0)-softness are shown to be local properties and it is deduced that each point of a Dugundji space and of a quotient space of a locally compact group has a neighbourhood base of Dugundji spaces. Moreover, we prove that a space and its cone are Dugundji spaces simultaneously. (∞,∞)-soft group homomorphisms are isomorphisms. This follows from the result that a compact contractible group is trivial.</p> <p>A compact space is classified as an absolute extensor for m- to n-dimensional spaces or AE(m,n), if the constant map is (m,n)~soft.</p> <p>Equivalently, a compact space is an AE(m,n) if and only if every embedding into a compact space is (m,n)-hard. Haydon's paper entails coincidence of the notions of AE(0,0) and Dugundji space. Here an AE(0,0) (resp. AE(0,∞)) is surjectively characterized as (-1,0)-soft image of a generalized Cantor set (resp. Tychonoff cube). An in- jective characterization of the class of Poano spaces is given as metrizable AE(0,n), n (1≤n≤∞) being arbitrary.</p> <p>Using combinatorial techniques we obtain some results concerning the countable chain condition on spaces of probability measures and spaces of ⩽ n-element subsets.</p> |
| spellingShingle | Hoffmann, B On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title | On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title_full | On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title_fullStr | On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title_full_unstemmed | On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title_short | On a classification of surjections and embeddings : surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| title_sort | on a classification of surjections and embeddings surjective characterization of injective classes of compact spaces with applications to compact and locally compact groups |
| work_keys_str_mv | AT hoffmannb onaclassificationofsurjectionsandembeddingssurjectivecharacterizationofinjectiveclassesofcompactspaceswithapplicationstocompactandlocallycompactgroups |