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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: 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

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