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abelian topological group additive group axiom of countability basis relative bourhood columns compact and satisfy compact commutative group continuous function converges correspondence is continuous correspondence is preserved countability axiom countable basis countable set defined definition 2.9 denote exists a neighbourhood finite vector space finitely non-zero rows form a basis G into G give all possible group isomorphic group with operator Hausdorff space hood infinite basis infinite direct sum infinite matrices infinitely non-zero integers lemma Let G limit point linear subspace linear transformations matrix theory multiplication by elements neigh neighbour neighbourhood of zero neighbourhoods in G non-singular matrices noted open set operator ring possible orders preserved under addition principal diagonal principal ideal ring Proof real numbers ring homomorphic ring of homomorphisms ring of infinite ring of matrices ring of operators satisfying the second second axiom subring Suppose lim theorem 2.5 theory of infinite thesis tinuous whence zero contains zero homomorphism