Vol. 15, No. 2, 1965

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Some general properties of multi-valued functions

Raymond Earl Smithson

Vol. 15 (1965), No. 2, 681–703

The object is to determine what theorems for single-valued functions can be extended to which class of multi-valued functions. It is shown that an arc cannot be mapped onto a circle by a continuous, monotone multi-valued function when the image of each point is an arc. On the other hand, the arc can be mapped onto a nonlocally connected space by a monotone, continuous function such that the image of each point is an arc. Characterizations of nonalternating functions analogous to the results in the single-valued theory are obtained, and it is shown that an nonalternating semi-single-valued continuous function on a dendrite is monotone. An analog of the monotone light factorization theorem is obtained for semi-single-valued continuous functions.

Some other results are: an open continuous function with finite images maps a regular curve onto a regular curve, and a continuous function with finite images maps a locally connected, compact space onto a locally connected compact space.

Mathematical Subject Classification
Primary: 54.65
Received: 3 April 1963
Revised: 31 March 1964
Published: 1 June 1965
Raymond Earl Smithson