On two-to-one continuous functions

J. Mioduszewski

GDML_Books, (1961), p.

Résumé

CONTENTSIntroduction................................................................................................................................................................................3I. General properties of k-to-one functions on locally compact spaces1. Multi-valued functions Ф and ψ......................................................................................................................................... 62. The proof of (I.11)............................................................................................................................................................... 73. Some conditions for ψ to be discontinuous at x ≠ p................................................................................................... 84. Some properties of $f{-1}f$ on $\mathcal {X}$ .............................................................................................................. 95. Partial functions................................................................................................................................................................... 96. A converse problem............................................................................................................................................................ 9II. General properties of two-to-one functions on locally compact spaces1. Semicontinuous involution φ............................................................................................................................................ 122. Classification of points of discontinuity of φ.................................................................................................................. 133. WED-points of φ|X, where X is a closed subset of $\mathcal {X}$........................................................................... 134. The function $\tilde{φ}$ on V............................................................................................................................................. 145. The function φ considered on $\overline{WED(φ|X)}$................................................................................................ 15III. Behaviour of semicontinuous involution at Euclidean points1. Definitions............................................................................................................................................................................ 172. The function (φ|X), where X is the closure of a spherical (hemispherical) pseudo-neighbourhoodof a PE-point............................................................................................................................................................................. 173. The main theorem.............................................................................................................................................................. 194. Immediate consequences of Theorem 1....................................................................................................................... 20IV. Some applications to compact spaces1. Civin’s theorems................................................................................................................................................................. 232. Two-to-one continuous functions on graphs................................................................................................................ 243. Two-to-one functions on the closures of plane domains.......................................................................................... 264. Two-to-one functions on some class of irreducible continua.................................................................................... 285. The non-existence of two-to-one continuous functions on the indecomposable continuum$\mathcal {B}_0$...................................................................................................................................................................... 30V. Two-to-one continuous functions on locally compact manifolds1. The function $\tilde{φ}$ on $\mathcal {X}$ — SED(φ).................................................................................................. 332. The strong accessibility of points of SED(φ)................................................................................................................. 343. The one-dimensional case............................................................................................................................................... 364. The existence of two-to-one continuous functions on Euclidean spaces $E^n$ for n ≥ 2.................................. 39References.............................................................................................................................................................................. 42

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