## Certain local properties of topological spacesShow full item record

Title | Certain local properties of topological spaces |
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Author | Marrache, Nazem M. |

Date | 1968 |

Genre | Dissertation |

Degree | Doctor of Philosophy |

Abstract | A topological space X is said to be an L-space if for each point x of X there is a collection, H = (H_alpha: alpha in A), of open sets and a compact set G such that: (1) x not in cl(H_alpha) for a in A, (2) H covers X - G, and (3) H is locally finite at each point of X - G. A space X is said to be k-countable if for each point x of X there is a sequence, {U_n} from n=1 to infinity, of neighborhoods of x such that the Intersection from n=1 to infinity of cl(U_n) is compact. The notion of L-space has the distinction of being a simultaneous generalization of the concepts of locally compact space, first countable space and k-countable space. Ernest Michael introduced the notions of r-space and of q-space each of which is a generalization of the concepts of locally compact and first countable space. The purpose of this dissertation is to study the properties of these spaces and to present some relations which exist among them. Specifically, the following theorems are proven. Theorem. If a space X is both an L-space and a q-space then X is a k-space. Theorem. If a space X is a k-countable q-space then X is both an r-space and a k-space. Theorem. A k-countable q-space is preserved under an open mapping. Theorem. The product space of two k-countable q-spaces is a k-countable q-space. Theorem. If X is a pseudo-compact L-space then X is a k-space. Theorem. If X is a pseudo-compact k-countable space then X is both an r-space and a k-space. Further, we show that the notion of L-space is sufficient to insure the pseudo-compactness of the product of two pseudo-compact spaces. It should be mentioned that pseudo-compactness which appeared in the above results has some interaction with these local properties. Finally, some local properties are discussed in terms of properties of compactifications. |

Link | https://repository.tcu.edu/handle/116099117/33798 |

Department | Mathematics |

Advisor | Tamano, Hisahiro |

##### This item appears in the following Collection(s)

- Doctoral Dissertations [1473]

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