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On two-to-one continuous functions

Seria
Rozprawy Matematyczne tom/nr w serii: 24 wydano: 1961
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Warianty tytułu
Abstrakty
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CONTENTS
Introduction................................................................................................................................................................................3
I. General properties of k-to-one functions on locally compact spaces
1. Multi-valued functions Ф and ψ......................................................................................................................................... 6
2. The proof of (I.11)............................................................................................................................................................... 7
3. Some conditions for ψ to be discontinuous at x ≠ p................................................................................................... 8
4. Some properties of $f{-1}f$ on $\mathscr{X}$ .............................................................................................................. 9
5. Partial functions................................................................................................................................................................... 9
6. A converse problem............................................................................................................................................................ 9
II. General properties of two-to-one functions on locally compact spaces
1. Semicontinuous involution φ............................................................................................................................................ 12
2. Classification of points of discontinuity of φ.................................................................................................................. 13
3. WED-points of φ|X, where X is a closed subset of $\mathscr{X}$........................................................................... 13
4. The function $\tilde{φ}$ on V............................................................................................................................................. 14
5. The function φ considered on $\overline {WED(φ|X)}$................................................................................................ 15
III. Behaviour of semicontinuous involution at Euclidean points
1. Definitions............................................................................................................................................................................ 17
2. The function (φ|X), where X is the closure of a spherical (hemispherical) pseudo-neighbourhood
of a PE-point............................................................................................................................................................................. 17
3. The main theorem.............................................................................................................................................................. 19
4. Immediate consequences of Theorem 1....................................................................................................................... 20
IV. Some applications to compact spaces
1. Civin’s theorems................................................................................................................................................................. 23
2. Two-to-one continuous functions on graphs................................................................................................................ 24
3. Two-to-one functions on the closures of plane domains.......................................................................................... 26
4. Two-to-one functions on some class of irreducible continua.................................................................................... 28
5. The non-existence of two-to-one continuous functions on the indecomposable continuum
$\mathscr{B}_0$...................................................................................................................................................................... 30
V. Two-to-one continuous functions on locally compact manifolds
1. The function $\tilde{φ}$ on $\mathscr{X}$ — SED(φ).................................................................................................. 33
2. The strong accessibility of points of SED(φ)................................................................................................................. 34
3. The one-dimensional case............................................................................................................................................... 36
4. The existence of two-to-one continuous functions on Euclidean spaces $E^n$ for n ≥ 2.................................. 39
References.............................................................................................................................................................................. 42
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 24
Liczba stron
43
Liczba rozdzia³ów
Opis fizyczny
Rozprawy Matematyczne, Tom XXIV
Daty
wydano
1961
Twórcy
  • University of Wrocław
Bibliografia
  • [1] P. Aleksandroff und H. Hopf, Topologie, Berlin-Leipzig 1935.
  • [2] R. H. Bing, The Cartesian product of a certain non manifolds and a line is E4, Bull. Amer. Math. Soc. 64 (1958), p. 82-84.
  • [3] K. Borsuk and R. Molski, On a class of continuous mappings, Fund. Math. 45 (1958), p. 84-98.
  • [4] P. Civin, Two-to-one mappings of manifolds, Duke Math. Jour. 10 (1943), p. 49-57.
  • [5] P. W. Gilbert, n-to-one mappings on linear graphs, ibidem 9 (1942), p. 475-486.
  • [6] O. G. Harrold, Exactly (k, 1) transformations on connected linear graphs. Amer. Jour. Math. 62 (1940), p. 823-834.
  • [7] O. G. Harrold, The non-existence of a certain type of continuous transformations, Duke Math. Jour. 5 (1939), p. 789-793.
  • [8] B. von Kérékjartó, Vorlesungen über Topologie, Berlin 1923.
  • [9] B. Knaster, Un continu irréductible à décomposition continue en tranches, Fund. Math. 25 (1935), p. 568-577.
  • [10] C. Kuratowski, Topologie, I (1948) and II (1950), Warszawa-Wrocław.
  • [11] Venable Martin and J. H. Roberts, Two-to-one transformations on 2-manifolds, Trans. Amer. Math. Soc. 49 (1941), p. 1-17.
  • [12] J. Mioduszewski, Funkcje ciągłe o stałej krotności skończonej na odcinku i prostej, Prace Matem. 5 (1961), p. 79-93.
  • [13] H. M. A. Newman, Л theorem on periodic transformations of spaces. The Quart. Jour. Math., Oxford Series, 2 (1931), p. 1-8.
  • [14] J. H. Roberts, Two-to-one transformations, Duke Math. Jour. 6 (1940), p. 256-262.
  • [15] P. A. Smith, Transformations of finite period, II, Annals of Math. 40 1939, p. 690-711.
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