A solution to a problem of Eilenberg concerning dimension lowering mappings

Authors:
James Keesling and David C. Wilson

Journal:
Proc. Amer. Math. Soc. **86** (1982), 159-162

MSC:
Primary 54F45; Secondary 54C99

DOI:
https://doi.org/10.1090/S0002-9939-1982-0663888-4

MathSciNet review:
663888

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper describes a map from a metric space (having the same dimension at each of its points) onto a space such that with the property that for every closed set contained in . This answers a question posed by Eilenberg in 1936 in *The Scottish Book*. This question was answered with a more complicated example by Rubin, Schori, and Walsh.

**[1]**R. D. Anderson,*Some remarks on totally disconnected sections of monotone open mappings*, Bull. Acad. Polon. Sei. Cl. III**4**(1956), 329-330. MR**0081462 (18:407c)****[2]**R. H. Bing,*Higher-dimensional hereditarily indecomposable continua*, Trans. Amer. Math. Soc.**7**(1951), 267-273. MR**0043452 (13:265c)****[3]**P. Civin and J. H. Roberts,*Sections of continuous collections*, Bull. Amer. Math. Soc.**49**(1943), 142-143. MR**0007642 (4:172d)****[4]**J. Keesling,*Mappings and dimension in general metric spaces*, Pacific J. Math.**25**(1968), 277-288. MR**0227954 (37:3538)****[5]**-,*Open mappings and closed subsets of the domain in general metric spaces*, Proc. Amer. Math. Soc.**20**(1969) 238-245. MR**0261568 (41:6181)****[6]**-,*Mappings and dimension in metric spaces*, General Topology and Appl.**2**(1972), 181-192. MR**0307147 (46:6268)****[7]**J. L. Kelley,*Hyperspaces of a continuum*, Trans. Amer. Math. Soc.**52**(1942), 22-36. MR**0006505 (3:315b)****[8]**W. S. Mahavier,*Dimension raising subsets of mappings onto an arc*, preprint. MR**978731 (90a:54077)****[9]**L. Rubin, R. M. Schori and J. Walsh,*New dimension-theory techniques for constructing infinite-dimensional examples*, General Topology and Appl.**10**(1979), 93-102. MR**519716 (80e:54049)****[10]**S. M. Ulam,*The Scottish book*, Los Alamos Scientific Laboratory Monograph LA-6832, 1957.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54F45,
54C99

Retrieve articles in all journals with MSC: 54F45, 54C99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0663888-4

Keywords:
Dimension,
mapping,
monotone mapping,
cell-like mapping,
dimension lowering mapping

Article copyright:
© Copyright 1982
American Mathematical Society