Vol. 45, No. 1, 1973

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ISSN: 0030-8730
Dendrites, dimension, and the inverse arc function

John Jobe

Vol. 45 (1973), No. 1, 245–256

In this paper, the concept of an inverse arc function is introduced. An inverse arc function f is a function such that for each arc L in the range of f, there exists an arc L1 in the domain of f such that f(L1) = L. It is proved that a dendrite D is the continuous image of an inverse arc function f with domain an arc L if and only if D has only a finite number of endpoints. Other results are obtained telling what dendrites can be ranges of continuous inverse arc functions having dendrites as domains.

The dimension raising ability of a continuous inverse arc function whose domain is a dendrite is questioned. It is proved that if D is a dendrite with only a countable number of endpoints, then there does not exist a continuous inverse arc function f with domain D such that dimf(D) 2. If a dendrite D has uncountably many endpoints, then the question is left unanswered.

Mathematical Subject Classification 2000
Primary: 54F50
Received: 7 December 1971
Revised: 5 May 1972
Published: 1 March 1973
John Jobe