Open maps of chainable continua

Author:
Ira Rosenholtz

Journal:
Proc. Amer. Math. Soc. **42** (1974), 258-264

MSC:
Primary 54F20; Secondary 54C10

DOI:
https://doi.org/10.1090/S0002-9939-1974-0331346-8

MathSciNet review:
0331346

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Abstract | References | Similar Articles | Additional Information

Abstract: It is apparently ``well known'' that the image of the closed unit interval under an open map is homeomorphic to the closed unit interval (see [**13**], [**11**], and [**15**]). In this paper, we generalize this result to chainable continua. In particular, the fact that the open continuous image of a chainable continuum is also chainable is proved, answering a question of A. Lelek (see [**10**]). This fact, as well as its proof, implies that the open continuous image of the pseudo-arc is also a pseudo-arc. An additional corollary (of the proof) is that a local homeomorphism of a chainable continuum is actually a homeomorphism. The proofs are all very elementary.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0331346-8

Keywords:
Open maps,
chainable continua,
local homeomorphisms,
pseudo-arc

Article copyright:
© Copyright 1974
American Mathematical Society