By Price

A slightly lovely little e-book, written within the kind of a textual content yet prone to be learn easily for excitement, during which the writer (Professor Emeritus of arithmetic on the U. of Kansas) explores the analog of the speculation of services of a posh variable which comes into being whilst the complexes are re

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**Example text**

1 ~ I}. It is a compact multiplicative semigroup, = {A 1 >.. € a:, and its group of invertible elements is ID' = ~. We remark here that every Hausdorff topological space can be 56 Chap. 2 TOPOLOGICAL GROUPS AND SEMI GROUPS endowed with a structure of topological semigroup by choosing an 0 and setting st = 0 for all sand t. this respect groups differ sharply from general semigroups. arbitrary point Exercise. topolog~cal A cannot be endowed ~egment w~th a ~t~uctu~e In 06 g~oup. A profound connection exists between compact semi groups and compact groups.

Complex or real) Banach space. Example 5. The group of automorphisms of a Banach space, endowed with the uniform topology (in GL(n,lR)). Example 6. algebra particular, GL(n,~) and The group of all invertible elements of a Banach A (with the topology induced from A). One of the important constructions in group theory, the direct product, admits a canonical topology. t x F, where additive group nal torus ~. F is any finite Abelian group. t particular x F g40up~ ; in is, by the classical structure theorem, the general form of finitely-generated Abelian groups).

Let D em is open, p + a is not necessarily Hausdorff. W be the subgroup of rational numbers in m. D t- ¢ , a E W belong to with topology on G/r m/w and for each D, then p D E D = m. all the numbers Therefore, the is anti-discrete. In contrast with this example we have the following THEOREM. 06 ed, ~hen G/r ~6 Chap. 2 TOPOLOGICAL GROUPS AND SEMIGROUPS 48 PROOF. , gl g2 E f. Since the complement of f in G is open, it follows from the continuity of the operation of left division that admit neighborhoods Nl and N2 , respectively, such But then [(N l rJ -1 (N 2 rJ 1 n f = ~, whence Nl f n N2f ~.