2-to-1 maps with hereditarily indecomposable images

Heath, Jo

doi:10.1090/S0002-9939-1991-1081696-5

$2$-to-$1$ maps with hereditarily indecomposable images
HTML articles powered by AMS MathViewer

by Jo Heath

Proc. Amer. Math. Soc. 113 (1991), 839-846

DOI: https://doi.org/10.1090/S0002-9939-1991-1081696-5

PDF | Request permission

Abstract:

It is shown that there is no $2$-to-$1$ map from the pseudoarc, or any treelike continuum, onto a hereditarily indecomposable continuum, and that no hereditarily indecomposable treelike continuum can be the image of a $2$-to-$1$ map.

References

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249. MR 220249, DOI 10.4064/fm-60-3-241-249

Sur quelques proprietes des transformations localement homeomorphes

V24

Jo Heath, Tree-like continua and exactly $k$-to-$1$ functions, Proc. Amer. Math. Soc. 105 (1989), no. 3, 765–772. MR 936775, DOI 10.1090/S0002-9939-1989-0936775-8
Jo Heath, The structure of (exactly) $2$-to-$1$ maps on metric compacta, Proc. Amer. Math. Soc. 110 (1990), no. 2, 549–555. MR 1013970, DOI 10.1090/S0002-9939-1990-1013970-1

Four key questions in the theory of

-to-

maps

to

maps on hereditarily indecomposable metric continua

Casimir Kuratowski, Topologie. I. Espaces Métrisables, Espaces Complets, Monografie Matematyczne, Vol. 20, Państwowe Wydawnictwo Naukowe (PWN), Warszawa-Wrocław, 1948 (French). 2d ed. MR 0028007
T. Bruce McLean, Confluent images of tree-like curves are tree-like, Duke Math. J. 39 (1972), 465–473. MR 305372
J. Mioduszewski, On two-to-one continuous functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 9 (1961), 129–132 (English, with Russian summary). MR 133120
Sam B. Nadler Jr. and L. E. Ward Jr., Concerning exactly $(n,\,1)$ images of continua, Proc. Amer. Math. Soc. 87 (1983), no. 2, 351–354. MR 681847, DOI 10.1090/S0002-9939-1983-0681847-3
Ira Rosenholtz, Open maps of chainable continua, Proc. Amer. Math. Soc. 42 (1974), 258–264. MR 331346, DOI 10.1090/S0002-9939-1974-0331346-8