A counterexample concerning almost continuous functions
| Autor: | W. F. Lindgren, S. A. Naimpally, W. N. Hunsaker, Larry L. Herrington |
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| Rok vydání: | 1974 |
| Předmět: | |
| Zdroj: | Proceedings of the American Mathematical Society. 43:475 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/s0002-9939-1974-0331306-7 |
| Popis: | An example is constructed of a function which is almost continuous in the sense of Singal and Singal but not in the sense of Stallings. Let X be the set of real numbers with the topology 7 consisting of the usual open sets together with the sets of the form UnD, where U is an open set in the usual topology and D the set of all irrational numbers. Letf: [0, 1]-+(X, 7) be defined byf(x)=x. Thenf is almost continuous in the sense of Singal and Singal (and also in the sense of Husain3). Since the only continuous functions on [0, 1]-+(X, 7) are the constant functions (Steen and Seeback [3, p. 89]), f is not almost continuous in the sense of Stallings. This answers an open problem recently posed by Long and Carnahan r21. REFERENCES 1. H. Blumberg, New properties of all real functions, Trans. Amer. Math. Soc. 24 (1922), 113-128. 2. P. E. Long and D. A. Carnahan, Comparing almost continuous functions, Proc. Amer. Math. Soc. 38 (1973), 413-418. 3. L. A. Steen and J. A. Seeback, Jr., Counterexamples in topology, Holt, Rinehart and Winston, New York, 1970. MR 42 #1040. UNIVERSITY OF ARKANSAS, FAYETTEVILLE, ARKANSAS 72701 SOUTHERN ILLINOIS UNIVERSITY, CARBONDALE, ILLINOIS 62901 SLIPPERY ROCK STATE COLLEGE, SLIPPERY ROCK, PENNSYLVANIA 16057 LAKEHEAD UNIVERSITY, THUNDER BAY, ONTARIO P7B SEI, CANADA Received by the editors July 12, 1973. AMS (MOS) subject classifications (1970). Primary 54C10. |
| Databáze: | OpenAIRE |
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