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For any regular space Z It is shown, 1) that the bounded-open topology T on C(Y,Z) is splitting and it is also the smallest jointly continuous topology whenever Y is locally bounded, 2) if Y is locally bounded or if X × Y is a boundedly generated space, then there is a natural bijection on C(X × Y,Z) onto C(X,(C(Y,Z),Teo) which is actually a homeomorphism with respect to the bounded-open topology on both function spaces, 3) The path components of (C(Y,Z),Teo) are exactly its homotopy classes whenever Y is boundedly generated, 4) The bounded-open topology Teo induces contravariant and covariant Homotopy preserving function-space functors. Further, 5) Teo reduces to the compact-open topology tco whenever the domain Y is regular; but in general, Teo is finer than Tco (assuming the domain is Hausdorff or the range is either Hausdorff or regular). 相似文献
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Panayotis Lambrinos 《manuscripta mathematica》1973,10(3):289-296
A topological boundedness notion is studied, which is proved to be productive. Classical theorems on compactness of Tychonoff, Alexander and Obreanu are generalized. A boundedness operator is defined and studied. Finally, a classification of all topological spaces is obtained according to boundedness criteria.The author is grateful to prof. N. Oeconomidis, who suggested the topic, for his continuous interest. 相似文献
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Panos Lambrinos 《manuscripta mathematica》1980,31(4):425-438
The new class of Boundedly generated topological spaces (or l-spaces) is defined and studied by topological methods. It is
shown that it is strictly broader than the class of (Hausdorff) compactly generated spaces (or k-spaces) and also that l-spaces
possess many of the nice properties of k-spaces e.g. they are closed under the formation of disjoint unions, quotients, direct
limits e.t.c. The topology of uniform convergence on boundeda is also studied and in general, it is shown to be strictly finer
than the compact-open topology on the space of continuous functions. 相似文献
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