every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact. Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal. Every locally compact group which is first-countable is metrisable as a topological group (i.e. can be given a left-invariant metric compatible with the topology) and complete. If furthermore the space is second-countable, the metric can be chosen to be proper. (See the article on topological groups.) In a Polish group G, the σ-algebra of Haar null sets satisfies the countable chain condition if and only if G is locally compact. Locally compact abelian groups For any locally compact abelian (LCA) group A, the group of continuous homomorphisms Hom(A, S1) from A to the circle group is again locally compact. Pontryagin duality asserts that this functor induces an equivalence of categories LCAop → LCA. This functor exchanges several properties of topological groups. For example, finite groups correspond to finite groups, compact groups correspond to discrete groups, and metrisable groups correspond to countable unions of compact groups (and vice versa in all statements). LCA groups form an exact category, with admissible monomorphisms being closed subgroups and admissible epimorphisms being topological quotient maps. It is therefore possible to consider the K-theory spectrum of this category. has shown that it measures the difference between the algebraic K-theory of Z and R, the integers and the reals, respectively, in the sense that there is a homotopy fiber sequence K(Z) → K(R) → K(LCA). See also References Further reading . Topological groups