On the images of a stratifiable space under certain open and psuedo-open mappings
Henry, David Michael
Henry, David Michael
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1970
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Abstract
A space is stratifiable if to each open set U there corresponds a countable collection {U_n} of open sets satisfying the following properties: (a) U-bar_n is a subset of U, (b) Union from n=1 to infinity of U_n = U, and (c) if U and V are open sets with U subset of V, then U_n is a subset of V_n for each n. If the definition is weakened by requiring that each U_n be arbitrary rather than open, then a space satisfying the three previous conditions is said to be semi-stratifiable. The class of stratifiable spaces lies strictly between the classes of metric and perfectly paracompact spaces, while a first countable semi-stratifiable space is semi-metrizable, and conversely. It is the purpose of this dissertation to investigate the images of a space X, which is usually assumed to be stratifiable, under certain types of open and pseudo-open mappings. In addition, several local properties in spaces where points are G_delta-sets are studied. Call a mapping f:X to Y compact if f^-1(y) is compact for each y in Y, cs-compact if f^-1(Z) is compact for each convergent sequence Z subset of Y, and proper if f^-1(C) is compact for each compact C subset of Y. Also, f is pseudo-open if for each y in Y and neighborhood K of f^-1(y), y in int[f(K)]. In view of these definitions, the following theorems have been established for Hausdorff spaces. Theorem: The image of a stratifiable space under a pseudo-open compact mapping is a semi-stratifiable space. Theorem: The image of a stratifiable Frechet space under an open cs-compact mapping is a paracompact semi-stratifiable Frechet space satisfying an additional weak convergence property. Theorem: The image of a first countable stratifiable space under an open proper mapping is a first countable stratifiable space. The previous theorem is proved without appealing to the fact that the proper mapping must also be closed. Furthermore, if the compact mapping in the first theorem is sharpened by requiring that it be finite-to-one, then we obtain the following result. Theorem: The image of a semi-stratifiable space under a pseudo-open finite-to-one mapping is a semi-stratifiable space. A space X is said to be of countable type if for each compact set C there exists a compact set K having a countable basis and such that C is a subset of K. Similarly, X is of point-countable type if each point of X is contained in a compact set which has a countable basis. Theorem: Let X be a Hausdorff space of countable type and assume that each compact set has a countable collection of neighborhoods whose intersection is that set. That is, each compact set is a G_delta-set. Then each compact set has a countable base. Theorem: A necessary and sufficient condition that a Hausdorff space be first countable is that it be of point-countable type and that each point be a G_delta-set.
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Metric spaces
Topology
Topology
Research Projects
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Dissertation
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iv, 41 leaves, bound
Department
Mathematics