Characteristic Classes and the Cohomology of Finite Groups by Charles Benedict Thomas

By Charles Benedict Thomas

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With the usual topology, the additive group (R, +) and the multiplicative group (R× , ·) are locally compact Hausdorff groups; here R× := R {0} carries the induced topology. The set C of complex numbers can be identified with R2 ; thus we obtain the natural topology on C. Again, the additive group (C, +) and the multiplicative group (C× , ·) are locally compact Hausdorff groups. If R is a topological ring, then the ring R n×n of all n × n matrices with entries in R is a topological ring, if it is endowed with the product topology (here 2 we identify R n×n with R (n ) ).

From now on, we use multiplicative notation for our groups, unless stated otherwise. Let G be a group. For elements x, y ∈ G and subsets X, Y ⊆ G we write Xy := {xy | x ∈ X}, xY := {xy | y ∈ Y } and XY := {xy | x ∈ X, y ∈ Y } = x∈X xY = y∈Y Xy. 10 implies that Xy, yX, XY and Y X are open for each y ∈ G and each Y ⊆ G. Similarly, if Y is closed in G, then Y x and xY are closed. However, XY need not be closed, even if both X and Y are closed. 16 Example. We consider the group of real numbers, written additively, with its usual topology.

Let X be a locally compact Hausdorff space. Then X is totally ✷ disconnected if, and only if ind X = 0. 18 Sum Theorem for small inductive dimension. Let X be a separable metrizable space. If (An )n∈N is a countable family of subsets of X then ind n∈N An = supn∈N ind An . Proof. See, for instance, [42], p. 14. 19 Theorem. For every natural number n, we have ind Rn = n. Proof. See [30], Th. 10. 20 Lemma. Let X be a nonempty regular topological space. (a) For every subspace Y of X, we have ind Y ≤ ind X.

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