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ON SEMICONNECTED MAPPINGS OF TOPOLOGICAL SPACES JOHN JONES, JR. 1. Introduction. Let (X, 11) and (Y, V) denote topological spaces as in Kelly [l]. A mapping/ of (X, 1L) into (Y, V) is said to be connected if and only if it maps connected subsets of (X, Ii) into connected subsets of (F, TJ). W. J. Pervin and N. Levine [3] and T. Tanaka [4] recently considered connected mappings of Hausdorff spaces (X, It) into (F, V). A mapping/of (X, Tl) into (F, TJ) is semiconnected if/"1(A) is a closed and connected set in (X, 1L) whenever A is a closed and connected set in (Y, V). A mapping/ is bi-semiconnected if and only if/and/-1 are each semiconnected. Using the definition of G. T. Whyburn [5] a connected T+space (X, It) is said to be semilocally connected (s.l.c.) at xEX if and only if there exists a local open base at xEX such that A"\Fhas only a finite number of components, where Vis any element of the local open base at x. Since continuous mappings are special cases of connected mappings it is of interest to know what conditions must be placed upon a given mapping or upon the topological spaces (X, It), (Y, V) in order to conclude that a given mapping/ is continuous or is a homeomorphism. Examples of connected mappings which are not continuous are given by C Kuratowski [2] and Pervin and Levine [3]. 2. Results. Theorem 1. Let / be a one-to-one onto semiconnected mapping 0/ a topological space (X, It) to a semilocally-connected topological T2 space (Y, V), then/is continuous. Let B be an open set in F, and /~1(B)=ACX. Choose a point xEA and let/(x) =yEB. Since (Y, V) is semilocally-connected there exists an open set BVQB containing y and Y\By consists of a finite number of distinct components. Let these components be designated by Bi, B2, B3, ■ • ■ , B„. Then Bv= F\U"=1 B,, where each B( is con- nected and closed. Let Ai=/-1(Bi), for i=l, 2, ■ ■ ■, n. Each At is closed and connected since/ is semiconnected. Now either x belongs to the closure of some Aj or it does not. Suppose that x belongs to the closure of some Aj, for some/= 1, 2, • • • , «. Now Aj\Jx is closed and connected since/was a semiconnected mapping. Thus /(A/Ux) =Bj Received by the editors October 20, 1966 and, in revised form, December 15, 1966. 174 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ONsemiconnected mappings of topological spaces 175 which is connected. So yEBj, but this is impossible since Bj is a component of Y\By which does not contain y. So x cannot belong to the closure of Aj for any j= 1, 2, 3, ■ ■ ■, n. There exists an open set 0i = A\cl At such that xEOit i=l, 2, 3, • ■ ■ , n. Now Gi = n?_1G,- is an open set in AT containing x. Also fAd =Bi, and fid A ,)^Bi. So we have F\/(cl A{)Q Y\Bt. Also KOx)= /(n Oi) = n fm = n /(z\ci a* \ i=i / i=i ,=i = fi [F\/(cl Ai)] C fi [Y\Bi] = Y\ fi 5.- = Bv. t=l «=1 Since / is a one-to-one an Ox such that fiOx)QBy, onto mapping t=l and for any By there exists f is a continuous mapping of (A, CU) into (F, =0). Theorem 2. Le2 (A, It), (F, 13) 6e semilocally-connected and AT2 topological spaces and f be a one-to-one bi-semiconnected It) onto (F, 13), then f is a homeomorphism. mapping of (A, According to Theorem 1 above / is continuous, and applying the same type argument as used in the proof of Theorem 1, we see that f~l is a one-to-one onto continuous mapping of (F, V) to (A, It). Hence / is a homeomorphism. The author wishes to acknowledge the helpful comments of the referee. References 1. J. L. Kelly, General topology, Van Nostrand, New York, 1955. 2. C. Kuratowski, Topologie. II, Hafner, Warsaw, 1952. 3. W. J. Pervin and N. Levine, Connected mappings of Hausdorff spaces, Proc. Amer. Math. Soc. 11 (1958), 488-496. 4. T. Tanaka, On the family of connected subsets and the topology of space, J. Math. Soc. Japan 7 (1955),389-393. 5. G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Pub]., Vol. 28, Amer. Math Soc., Providence, R. I., 1942. George Washington University License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use