The space X, called totally disconnected if every two distinct points x, y is a member of X have disjoint neighborhoods which are both open and closed is discussed. It is proved that if G is an LCA group with dim G greater than or = n+1, then G contains a totally disconnected subgroup H with dim H greater than or = n. If G is metrizable then H can be constructed to be a Borel subset of G.
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