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If the parameters of a Lie group vary over a closed interval, them the Lie group is said to be compact. Every representation of a compact group is equivalent to a unitary ...

The compact-open topology is a common topology used on function spaces. Suppose X and Y are topological spaces and C(X,Y) is the set of continuous maps from f:X->Y. The ...

A sigma-compact topological space is a topological space that is the union of countably many compact subsets.

A compact manifold without boundary.

A product space product_(i in I)X_i is compact iff X_i is compact for all i in I. In other words, the topological product of any number of compact spaces is compact. In ...

A topological space X has a one-point compactification if and only if it is locally compact. To see a part of this, assume Y is compact, y in Y, X=Y\{y} and x in X. Let C be ...

Let G be a locally compact Abelian group. Let G^* be the group of all continuous homeomorphisms G->R/Z, in the compact open topology. Then G^* is also a locally compact ...

The Betti numbers of a compact orientable n-manifold satisfy the relation b_i=b_(n-i).

A Radon measure is a Borel measure that is finite on compact sets.

A group action of a topological group G on a topological space X is said to be a proper group action if the mapping G×X->X×X(g,x)|->(gx,x) is a proper map, i.e., inverses of ...