Vol. 45, No. 1, 1973

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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