By Gao J.

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Is compact, so that the canonical projection W G1 ! G1 / ! G2 is proper. '/ ! G1 / is an isomorphism, in particular is proper. It follows that the composition ' D j ı ı is proper. 6(1), the hypothesis “G1 is -compact” cannot be deleted, as id the example Rdis ! R shows, where Rdis stands for the group R with the discrete topology. 7. On a -compact LC-group, a continuous left-invariant metric is compatible. 1. Let X be a topological space. 84]. A topological space is totally disconnected if every point in it is its own connected component.

Let K be a non-discrete complete valued field and E ¤ f0g a Hausdorff topological vector space over K; if E is locally compact, then K is a non-discrete locally compact field and E is finite-dimensional [BEVT, chapitre 1, 2, no 4]. Recall also that, with the weak topology, an infinitedimensional Hilbert space is non-metrizable; its unit ball is metrizable if and only if the space is separable [Halm–74, Problems 18, 19, and 21]. A. Markov. 8 in [HeRo–63], Abels’ article [Abel–720 ], an introduction to free topological groups [Todo–97, Section 25], and an extended review [Sipa–03].

The finite groups K=N and the natural epimorphisms constitue an inverse system, its inverse limit is isomorphic to K, and the group K is the inverse limit of this inverse system. d; Z=pn Z/ n n are profinite groups. (3) Let K be a local field, G an algebraic group defined over K, and G the group of its K-points. K/ of n-by-n matrices over K; it follows that G has a natural topology T inherited from the topology of K, and G with T is a totally disconnected LC-group. When K is Qp or a finite extension of Qp , this topology is sometimes called the p-adic topology on G.

### A C^2 Finite Element and Interpolation by Gao J.

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