Vol. 47, No. 2, 1973

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Fixed point sets

L. E. Ward

Vol. 47 (1973), No. 2, 553–565

The principal results are the following. If M is a metric space homeomorphic to a subset of a real linear space which is star-shaped with respect to an element p, or if M is homeomorphic to an arcwise connected subspace of a dendroid which is smooth at a point p, then each closed subset of M which contains p is the fixed point set of a continuous mapping of M. If M is a continuum having Property W (this is a class of Peano continua containing the local dendrites and the continua containing no continuum of condensation) then each nonempty closed subset of M is a fixed point set. It is shown that a subset K of a dendrite is the fixed point set of a continuous surjection if and only if the complement of K is not homeomorphic to [0,).

Mathematical Subject Classification 2000
Primary: 54H25
Secondary: 54F50
Received: 17 May 1972
Published: 1 August 1973
L. E. Ward