A continuous circle of pseudo-arcs

filling up the annulus

Author:
Janusz R. Prajs

Journal:
Trans. Amer. Math. Soc. **352** (2000), 1743-1757

MSC (1991):
Primary 54F15; Secondary 54F50, 54B15

DOI:
https://doi.org/10.1090/S0002-9947-99-02330-2

Published electronically:
July 1, 1999

MathSciNet review:
1608498

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continuous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, ``a selectible circle of pseudo-arcs", and answers some questions of Lewis.

**1.**R.D. Anderson,*On collections of pseudo-arcs*, Abstract 337t, Bull. Amer. Math. Soc. 56(1950), 350.**2.**R. D. Anderson and G. Choquet,*A plane continuum no two of whose nondegenerate subcontinua are homeomorphic: an application of inverse limits*, Proc. Amer. Math. Soc. 10(1959), 347-353. MR**21:3819****3.**R. H. Bing,*Concerning hereditarily indecomposable continua*, Pacific J. Math. 1(1951), 43-51. MR**13:265b****4.**R. H. Bing and F. B. Jones,*Another homogeneous plane continuum*, Trans. Amer. Math. Soc. 90(1959), 171-192. MR**20:7251****5.**M. Brown,*Continuous collections of higher dimensional continua*, Ph. D. Thesis, University of Wisconsin, 1958.**6.**B. Knaster,*Un continu dont tout sous-continu est indecomposable*, Fund. Math. 3(1922), 247-286.**7.**W. Lewis,*Continuous curves of pseudo-arcs*, Houston J. Math. 11(1985), 91-99. MR**86e:54038****8.**W. Lewis,*The pseudo-arc*, Contemporary Math. 117(1991), 103-123. MR**92h:54046****9.**W. Lewis and J. J. Walsh,*A continuous decomposition of the plane into pseudo-arcs*, Houston J. Math. 4(1978), 209-222. MR**58:2750****10.***List of the works achieved in the field of mathematics and sciences in Poland during the German occupation, 1939-1945*, Polish Academy of Sciences and Letters, Kraków 1947. MR**10:276e****11.**E. E. Moise,*An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua*, Trans. Amer. Math. Soc. 63(1948), 581-594. MR**10:56i****12.**R. L. Moore,*Concerning upper semi-continuous collections of continua*, Trans. Amer. Math. Soc. 27(1925), 416-428.**13.**R. B. Sher,*Realizing cell-like maps in Euclidean space*, General Topology and its Appl. 2(1972), 75-89. MR**46:2683**

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Additional Information

**Janusz R. Prajs**

Affiliation:
Institute of Mathematics, Opole University, ul. Oleska 48, 45-052 Opole, Poland

Email:
jrprajs@math.uni.opole.pl

DOI:
https://doi.org/10.1090/S0002-9947-99-02330-2

Keywords:
continuous decomposition,
continuum,
plane,
pseudo-arc

Received by editor(s):
May 25, 1994

Received by editor(s) in revised form:
February 3, 1998

Published electronically:
July 1, 1999

Dedicated:
To the memory of Professor Bronisław Knaster

Article copyright:
© Copyright 2000
American Mathematical Society