| Summary: | Includes bibliography. === This thesis has its origin in a ten minute paper presented at the British Mathematical Colloquium in London in 1966. At this reading, a certain class of spaces was introduced by Mr. D. H. Fremlin, who furthermore, cited interesting compactness properties pertaining to these spaces. This work shall be concerned, foremostly, with the provision of proofs for the statements made by Mr. D. H. Fremlin in 1966, and with demonstrations of how these purely topological results have applications to linear topological spaces, with the emphasis placed on those which are locally convex. Our investigations of the 'Fremlin' spaces, and the search for known spaces which satisfy the 'Fremlin' conditions has led to a concentration on the separable and Frechet spaces. These are discussed in chapter 11. On reading this thesis, it will become apparent that an underlying theme is that of weak compactness. We have therefore taken the liberty, of concluding with a chapter which gives various characterisations of this. Each chapter will introduce its own subject matter. New results and different proofs of existing theorems have been obtained.
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