Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic |
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Authors: | Mariagnese Giusto Alberto Marcone |
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Affiliation: | (1) via Loreto Vecchia 9/10/A, I-17100 Savona, Italy (e-mail: giusto@dm.unito.it) , IT;(2) Dip. di Matematica, Università di Torino, via Carlo Alberto 10, I-10123 Torino, Italy (e-mail: marcone@dm.unito.it) , IT |
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Abstract: | We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ; the statement “every Lebesgue space is Atsuji” is provable in ; the statement “every Atsuji space is Lebesgue” is provable in . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to . Received: February 2, 1996 |
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Keywords: | Mathematics Subject Classification:03F35 |
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