Książka - szczegóły

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