Minimal Surfaces Spanning Closed Manifolds.

with respect to the property of being the product of continua of type P, where a continuum C is said to be of type P provided the closure of the complement of C consists of a finite number of components Di, each Di intersecting C in a single point pi, and where T(p,) e (D, pi). It is then shown that N is a subset of an F-set, and finally that N contains a continuum 7r such that T(7r) Dir. As results of 3.1 we obtain THEOREM 3.2: If T(M) CM is a continuous transformation of a compact continuum M, there exists either a fixed point in M or an F-set F such that F. T(F) contains a non-degenerate continuum. THEOREM 3.3: If T(M)CM is a continuous transformation of a compact continuum M, there exists a compact subset R of an F-set F of M such that T(R) = R. THEOREM 3.4: If T(M)CM is a continuous transformation of a compact continuum M which carries each F-set into a subset of an F-set--if, for example, the inverse of no point separates an F-set in M,-then there exists an Fset F such that T(F)C F. In case every F-set is degenerate, we have from 3.4 the Scherrer fixed point theorem for dendrites.6 If T is a homeomorphism and Ml is locally connected, 3.4 is Ayres' theorem.7 'Presented to the Amer. Math. Soc., Dec., 1938. The paper in full was offered to Fundamenta Mathematicae for publication in June, 1939. 2 Compare with Kuratowski and Whyburn, Fund. Math., 16, 305-331 (1930), and Moore, R. L., Foundations of Point Set Theory, p. 72. 3 peM is regular if there exists an arbitrarily small neighborhood of p with a finite boundary. See Menger, Kurventheorie, p. 96. 4Whyburn, G. T., Trans. Amer. Math. Soc., 30, 597-609 (1925). 6 A set N C M is a retract of M if there exists a continuous transformation T(M) = N such that T is the identity transformation on N. See Borsuk, K., Fund. Math., 17, 155 (1931). 6 Math. Zeit., 25, 129 (1926). 7 Fund. Math., 16, 333-336 (1930).