A topological group G is known as a B(𝑎) group if every continuous and almost open homomorphism from G onto a Hausdorff group is open. The permanence properties of the category of B(𝑎) groups are investigated and an internal characterization of such groups is established. Extensions of the closed graph and open mapping theorem are proved, employing this and related categories of groups. A similar concept is defined for topological semigroups, and further extensions of the open mapping and closed graph theorem are proved for them. / Thesis / Doctor of Philosophy (PhD)

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