Embeddings of infinite-dimensional manifold pairs and remarks on stability and deficiency

Katsuro SAKAI

doi:10.2969/jmsj/02920261

Katsuro SAKAI

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JOURNAL FREE ACCESS

1977 Volume 29 Issue 2 Pages 261-280

DOI https://doi.org/10.2969/jmsj/02920261

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Published: 1977 Received: March 23, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: ABSTRACT Details: Right : In this paper, we treat of an E-manifold pair (M, N) with N a Z-set in M where E is an infinite-dimensional locally convex linear metric space which is homeomorphic to E^ω or E^ω_f. And we study the condition under which M can be embedded in E such that N is the topological boundary under the embedding (Anderson's Problem in [2]). Moreover we extend the results on topological stability and deficiency, the Homeomorphism Extension Theorem and the results in [18].

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. D. Anderson, On topological infinite deficiency, Michigan Math. J., 14 (1967), 365-383.
2) R. D. Anderson, Some open questions in infinite-dimensional topology, Proc. 3-rd Prague Top. Symp. 1971, 29-35.
3) R. D. Anderson ane R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 74 (1968), 771-792.
4) R. D. Anderson and J. D. McCharen, On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc., 25 (1970), 283-289.
5) R. Arens and J. Eelles, On embedding uniform and topological spaces, Pacific J. Math., 6 (1956), 397-403.
6) K. Borsuk, Theory of Retracts, Monografie Mat. Tom. 44, PWM Warszawa, 1966.
7) M. Brown, Locally flat embeddings of manifolds, Ann. of Math., 75 (1962), 331-342.
8) T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc., 154 (1971), 399-426.
9) T. A. Chapman, Deficiency in infinite-dimensional manifolds, General Topology and Appl., 1 (1971), 263-272.
10) W. H. Cutler, Negligible subsets of infinite-dimensional Fréchet manifolds, Proc. Amer. Math. Soc., 23 (1969), 668-675.
11) W.H. Cutler, Deficiency in F-manifolds, Proc. Amer. Math. Soc., 34 (1972), 260-266.
12) R. Geoghegan, and D. W. Henderson, Stable function spaces, Amer. J. Math., 95 (1973), 461-470.
13) D. W. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, General Topology and Appl., 1 (1971), 321-327.
14) D. W. Henderson and J. E. West, Triangulated infinite-dimensional manifolds, Bull. Amer. Math. Soc., 76 (1970), 655-660.
15) S.-T. Hu, Theory of Retracts, Wayne St. Univ. Press, 1965.
16) V. L. Klee, Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc., 74 (1953), 10-43.
17) E. Michael, Convex structures and continuous selections, Canad. J. Math., 11 (1959), 556-575.
18) K. Sakai, On embeddings of infinite-dimensional manifold pairs, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 12 (1974), 202-213.
19) K. Sakai, An embedding of l²-manifold pairs in l², J. Math. Soc. Japan, 27 (1975), 557-560.
20) K. Sakai, An embedding theorem of infinite-dimensional manifold pairs in the modelled space, to appear in Fund. Math.
21) R. M. Schori, Topological stability for infinite-dimensional manifolds, Compositio Math., 23 (1971), 87-100.
22) H. Torunczyk, Remarks on Anderson's paper “On topological infinite deficiency”. Fund. Math., 66 (1970), 393-401.
23) H. Toruilczyk, A short proof of Hausdorff's theorem on extending metrics, Fund, Math., 77 (1972), 191-193.
24) H. Tortunzyk, Absorute retracts as factors of normed linear spaces, Fund. Math., 86 (1974), 53-67.
25) H. Torunczyk, On Cartesian factors and the topological classification of linear metric spaces, Fund. Mth., 88 (1975), 71-86.

Right : [1] R. D. Anderson, On topological infinite deficiency, Michigan Math. J., 14 (1967), 365-383.
[2] R. D. Anderson, Some open questions in infinite-dimensional topology, Proc. 3-rd Prague Top. Symp. 1971, 29-35.
[3] R. D. Anderson ane R. H. Bing, A complete elementary proof that Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc., 74 (1968), 771-792.
[4] R. D. Anderson and J. D. McCharen, On extending homeomorphisms to Fréchet manifolds, Proc. Amer. Math. Soc., 25 (1970), 283-289.
[5] R. Arens and J. Eelles, On embedding uniform and topological spaces, Pacific J. Math., 6 (1956), 397-403.
[6] K. Borsuk, Theory of Retracts, Monografie Mat. Tom. 44, PWM Warszawa, 1966.
[7] M. Brown, Locally flat embeddings of manifolds, Ann. of Math., 75 (1962), 331-342.
[8] T. A. Chapman, Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. Amer. Math. Soc., 154 (1971), 399-426.
[9] T. A. Chapman, Deficiency in infinite-dimensional manifolds, General Topology and Appl., 1 (1971), 263-272.
[10] W. H. Cutler, Negligible subsets of infinite-dimensional Fréchet manifolds, Proc. Amer. Math. Soc., 23 (1969), 668-675.
[11] W. H. Cutler, Deficiency in F-manifolds, Proc. Amer. Math. Soc., 34 (1972), 260-266.
[12] R. Geoghegan, and D. W. Henderson, Stable function spaces, Amer. J. Math., 95 (1973), 461-470.
[13] D. W. Henderson, Corrections and extensions of two papers about infinite-dimensional manifolds, General Topology and Appl., 1 (1971), 321-327.
[14] D. W. Henderson and J. E. West, Triangulated infinite-dimensional manifolds, Bull. Amer. Math. Soc., 76 (1970), 655-660.
[15] S. -T. Hu, Theory of Retracts, Wayne St. Univ. Press, 1965.
[16] V. L. Klee, Jr., Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soc., 74 (1953), 10-43.
[17] E. Michael, Convex structures and continuous selections, Canad. J. Math., 11 (1959), 556-575.
[18] K. Sakai, On embeddings of infinite-dimensional manifold pairs, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A, 12 (1974), 202-213.
[19] K. Sakai, An embedding of l²-manifold pairs in l², J. Math. Soc. Japan, 27 (1975), 557-560.
[20] K. Sakai, An embedding theorem of infinite-dimensional manifold pairs in the modelled space, to appear in Fund. Math.
[21] R. M. Schori, Topological stability for infinite-dimensional manifolds, Compositio Math., 23 (1971), 87-100.
[22] H. Torunczyk, Remarks on Anderson's paper “On topological infinite deficiency” Fund. Math., 66 (1970), 393-401.
[23] H. Torunczyk, A short proof of Hausdorff's theorem on extending metrics, Fund, Math., 77 (1972), 191-193.
[24] H. Tortunzyk, Absorute retracts as factors of normed linear spaces, Fund. Math., 86 (1974), 53-67.
[25] H. Torunczyk, On Cartesian factors and the topological classification of linear metric spaces, Fund. Mth., 88 (1975), 71-86.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

In this paper, we treat of an E-manifold pair (M, N) with N a Z-set in M where E is an infinite-dimensional locally convex linear metric space which is homeomorphic to E^ω or E^ω_f. And we study the condition under which M can be embedded in E such that N is the topological boundary under the embedding (Anderson's Problem in [2]). Moreover we extend the results on topological stability and deficiency, the Homeomorphism Extension Theorem and the results in [18].

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