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1
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On two-to-one continuous functions

100%
Mioduszewski J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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
2

O inwolucjach generowanych przez funkcje dwukrotne na przestrzeniach euklidesowych

94%
Mioduszewski J.
Commentationes Mathematicae
|
1962
|
tom 7
|
nr 1
EN
The article contains no abstract
3
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On the necessary and sufficient conditions for the analytic function to be univalent or p-valent in the usual and in the generalized sense

83%
Mioduszewski J.
Annales Polonici Mathematici
|
1959-1960
|
tom 7
|
nr 2
127-133
4
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Acknowledgment to the paper: "Another proof that a chain of non-empty $H$-closed subspaces has a non-empty intersection'

71%
Mioduszewski J.
Colloquium Mathematicum
|
1971
|
tom 23
|
nr 2
323
5
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H-closed and extremally disconnected Hausdorff spaces

66%
Mioduszewski J., Rudolf L.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction.......................................................................................................................................................... 5 I. PRELIMINARIES.............................................................................................................................................. 7 § 1. The closures of open subsets in r. o.-equivalent topologies............................................................. 7 § 2. The r. o.-maximal topologies.................................................................................................................... 9 § 3. The H-closed maximal spaces................................................................................................................ 10 § 4. R. o.-equivalence of extensions............................................................................................................... 10 § 5. 0-continuous maps.................................................................................................................................... 11 § 6. The Henriksen-Jerison and skeletal maps........................................................................................... 13 II. H-CLOSED EXTENSIONS OF HAUSDORFF SPACES.................................................................................... 14 § 1. The set of 77-closed extensions of given Hausdorff space............................................................... 14 5 2. Proper maps................................................................................................................................................ 16 § 3. Decompositions of proper maps............................................................................................................ 18 § 4. An application to IT-closed extensions................................................................................................... 19 § 5. The case of compact-like spaces........................................................................................................... 22 § 6. The case of minimal Hausdorff spaces................................................................................................. 25 III. EXTREMALLY DISCONNECTED RESOLUTIONS OF HAUSDORFF SPACES................................. 26 § 1. The set of irreducible maps onto a given Hausdorff space X............................................................ 26 § 2. R. o.-minimal irreducible maps............................................................................................................... 30 § 3. Extremally disconnected resolutions...................................................................................................... 31 IV. COMMUTATION OF H-CLOSED EXTENSIONS AND E. D. RESOLUTIONS...................................... 35 § 1. Commutativity in a pullback diagram...................................................................................................... 35 § 2. Commutativity in a pushout diagram ..................................................................................................... 37 V. PROJECTIVE AND INJECTIVE HAUSDORFF SPACES......................................................................... 39 § 1. H-closed projective spaces. A definition and motivations.................................................................. 41 § 2. The case of compact-like spaces........................................................................................................... 42 § 3. Projectiveness for arbitrary H-closed spaces....................................................................................... 44 § 4. Projectiveness for arbitrary Hausdorff spaces...................................................................................... 45 § 5. Injective extremally disconnected spaces............................................................................................. 46 § 6. Injective Hausdorff spaces....................................................................................................................... 48 Bibliography......................................................................................................................................................... 51
6
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Conditions which ensure that a simple map does not raise dimension

60%
Dębski W., Mioduszewski J.
Colloquium Mathematicum
|
1992
|
tom 63
|
nr 2
173-185
EN
The present paper deals with those continuous maps from compacta into metric spaces which assume each value at most twice. Such maps are called here, after Borsuk and Molski (1958) and as in our previous paper (1990), simple. We investigate the possibility of decomposing a simple map into essential and elementary factors, and the so-called splitting property of simple maps which raise dimension. The aim is to get insight into the structure of those compacta which have the property that simple maps from them do not raise dimension. In what follows a map means a continuous map, unless explicitly stated otherwise. A space is, except in some general lemmas, understood to be metrizable. A compactum means a compact metric space.
7
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Everywhere oscillating functions, extension of the uniformization and homogeneity of the pseudo-arc

59%
Mioduszewski J.
Fundamenta Mathematicae
|
1964
|
tom 56
|
nr 2
131-155
8
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Exactly two-to-one maps from continua onto arc-continua

49%
Dębski W., Heath J., Mioduszewski J.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 2
113-126
EN
Continuing studies on 2-to-1 maps onto indecomposable continua having only arcs as proper non-degenerate subcontinua - called here arc-continua - we drop the hypothesis of tree-likeness, and we get some conditions on the arc-continuum image that force any 2-to-1 map to be a local homeomorphism. We show that any 2-to-1 map from a continuum onto a local Cantor bundle Y is either a local homeomorphism or a retraction if Y is orientable, and that it is a local homeomorphism if Y is not orientable.
9
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Exactly two-to-one maps from continua onto some tree-like continua

49%
Dębski W., Heath J., Mioduszewski J.
Fundamenta Mathematicae
|
1992
|
tom 141
|
nr 3
269-276
EN
It is known that no dendrite (Gottschalk 1947) and no hereditarily indecomposable tree-like continuum (J. Heath 1991) can be the image of a continuum under an exactly 2-to-1 (continuous) map. This paper enlarges the class of tree-like continua satisfying this property, namely to include those tree-like continua whose nondegenerate proper subcontinua are arcs. This includes all Knaster continua and Ingram continua. The conjecture that all tree-like continua have this property, stated by S. Nadler Jr. and L. E. Ward Jr. (1983), is still neither confirmed nor rejected.
10
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On certain estimations of coefficients of univalent analytic functions

48%
Mioduszewski J.
Annales Polonici Mathematici
|
1959-1960
|
tom 7
|
nr 2
135-140
11
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Sur l'accessibilité des points d'ensembles fermés dans les espaces euclidiens

48%
Mioduszewski J.
Fundamenta Mathematicae
|
1958
|
tom 45
|
nr 1
314-319
12
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On a quasi-ordering in the class of continuous mappings of a closed interval into itself

48%
Mioduszewski J.
Colloquium Mathematicum
|
1962
|
tom 9
|
nr 2
233-240
13
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Another proof that a chain of non-empty H-closed subspaces has a non-empty intersection

48%
Mioduszewski J.
Colloquium Mathematicum
|
1971
|
tom 23
|
nr 1
43-44
14
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Simple plane images of the Sierpiński triangular curve are nowhere dense

42%
Dębski W., Mioduszewski J.
Colloquium Mathematicum
|
1990
|
tom 59
|
nr 1
125-140
15
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On the domains of indetermination of analytic functions

42%
Mioduszewski J.
Annales Polonici Mathematici
|
1960
|
tom 8
|
nr 3
241-248
16
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On projective spaces and resolutions in categories of completely regular spaces

36%
Mioduszewski J., Rudolf L.
Colloquium Mathematicum
|
1967
|
tom 18
|
nr 1
185-196
17
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Remarks on fixed point theorem for inverse limit spaces

36%
Mioduszewski J., Rochowski M.
Colloquium Mathematicum
|
1962
|
tom 9
|
nr 1
67-71
18
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Remarks on Baire theorem for H-closed spaces

36%
Mioduszewski J.
Colloquium Mathematicum
|
1971
|
tom 23
|
nr 1
39-41
19
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Direct limits of topological spaces and groups

36%
Mioduszewski J.
Colloquium Mathematicum
|
1964
|
tom 12
|
nr 1
15-22
20
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Compact Hausdorff spaces with two open sets

36%
Mioduszewski J.
Colloquium Mathematicum
|
1978
|
tom 39
|
nr 1
35-40
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