Weak and Weak Topologies

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The purpose of this project is to endow E with a smallest topology such that the maps фi,I Є I are continuous. We construct a topology r on E with minimum open sets which makes { фi }iЄI, continuous, where E is a non-empty set,{Yi}iЄI a family of topological spaces and, , continuous maps that take open seUts of E into open sets in Yi. фi: E → Yi Let ωi Є Yi. Therefore when ωi are open in Yi, i Є I, the set фi -1 (ωi) constitute a family of subsets of E which are necessarily open subsets in the topology τ; and we denote this family by U = {Ui}iЄI , where Ui = фi -1 (ωi) for some i Є I. The desired topology is constructed by finding the finite intersections of members of U and then arbitrary unions.
A thesis submitted to the Board of Postgraduate Studies, Kwame Nkrumah University of Science and Technology (KNUST), Kumasi, Ghana, in partial fulfilment of the requirements for the award of’ the degree of Master of Science (Msc.) in Mathematics, 2005