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Sequential space methods

University of British Columbia

The class of sequential spaces and its successive smaller subclasses, the Fréchet spaces and the first-countable spaces, have topologies which are completely specified by their convergent sequences. Because sequences have many advantages over nets, these topological spaces are of interest. Special attention is paid to those properties of first-countable spaces which can or cannot be generalized to Fréchet or sequential spaces. For example, countable compactness and sequential compactness are equivalent in the larger class of sequential spaces. On the other hand, a Fréchet space with unique sequential limits need not be Hausdorff, and there is a product of two Fréchet spaces which is not sequential. Some of the more difficult problems are connected with products. The topological product of an arbitrary sequential space and a T₃ (regular and T₁) sequential space X is sequential if and only if X is locally countably compact. There are also several results which demonstrate the non-productive nature of Fréchet spaces. The sequential spaces and the Fréchet spaces are precisely the quotients and continuous pseudo-open images, respectively, of either (ordered) metric spaces or (ordered) first-countable spaces. These characterizations follow from those of the generalized sequential spaces and the generalized Fréchet spaces. The notions of convergence subbasis and convergence basis play an important role here. Quotient spaces are characterized in terms of convergence subbases, and continuous pseudo-open images in terms of convergence bases. The equivalence of hereditarily quotient maps The class of sequential spaces and its successive smaller subclasses, the Fréchet spaces and the first-countable spaces, have topologies which are completely specified by their convergent sequences. Because sequences have many advantages over nets, these topological spaces are of interest. Special attention is paid to those properties of first-countable spaces which can or cannot be generalized to Fréchet or sequential spaces. For example, countable compactness and sequential compactness are equivalent in the larger class of sequential spaces. On the other hand, a Fréchet space with unique sequential limits need not be Hausdorff, and there is a product of two Fréchet spaces which is not sequential. Some of the more difficult problems are connected with products. The topological product of an arbitrary sequential space and a T₃ (regular and T₁) sequential space X is sequential if and only if X is locally countably compact. There are also several results which demonstrate the non-productive nature of Fréchet spaces. The sequential spaces and the Fréchet spaces are precisely the quotients and continuous pseudo-open images, respectively, of either (ordered) metric spaces or (ordered) first-countable spaces. These characterizations follow from those of the generalized sequential spaces and the generalized Fréchet spaces. The notions of convergence subbasis and convergence basis play an important role here. Quotient spaces are characterized in terms of conver-gence subbases, and continuous pseudo-open images in terms of convergence bases. The equivalence of hereditarily quotient maps and continuous pseudo-open maps implies the latter result. and continuous pseudo-open maps implies the latter result.

Text

2011-04-12

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http://hdl.handle.net/2429/33569

Mathematics

University of British Columbia

Graduate