One calls topological group all group ( G , *) provided with a topology satisfying the following conditions:

• the application ($x,\; there)\; \backslash \; to\; X\; *\; y$ is continuous
• the application $x\; \backslash \; to\; x^\; \{-\; 1\}$ is continuous

Basic examples

• the additive group $\backslash \; mathbb\; \{R\}\; \backslash ,$

One can show that a sub-group of $\backslash \; mathbb\; \{R\}\; \backslash ,$ dense is either , or form $a\; \backslash \; mathbb\; \{Z\}\; \backslash ,$, for single $has\; \backslash \; Ge\; 0\; \backslash ,$.

• the circle $S^1\; \backslash ,$, which can be condidéré as the group

multiplicative of the complex numbers of module $1\; \backslash ,$ or as the group of rotations of center fixed in an Euclidean plan. Any sub-group $S^1\; \backslash ,$ either is finished or dense.

• an example more sophisticated is $\backslash \; big\; (\backslash \; mathbb\; \{Z\}\; /2\; \backslash \; mathbb\; \{Z\}\; \backslash \; big)\; ^\; \backslash \; mathbb\; \{NR\}\; \backslash ,$

This group is homeomorphic with the Ensemble of Cantor. To see it, one needs the concept of produces infinite topological spaces.

Some general properties

• In a topological group, the translations

$X\; \backslash \; mapsto\; a*x\; \backslash ,$ and $X\; \backslash \; mapsto\; x*a\; \backslash ,$ are homeomorphisms.

• topology is determined by

the data of the vicinities of the neutral element.

• a topological group is separate if and only if

$\backslash \; \{E\; \backslash \}\; \backslash ,$ is closed in $G\; \backslash ,$, or, which returns to same, if all not is a closed part. The condition is obviously necessary. To see that it is sufficient, let us note that because continuity of the multiplication, for all open $U\; \backslash ,$ containing the neutral element, there exists about it another, that we will note $V\; \backslash ,$, such as $V*V\; \backslash \; subset\; U\; \backslash ,$. Even if it means to replace $V\; \backslash ,$ by $V\; \backslash \; course\; V^\; \{-\; 1\}\; \backslash ,$, one can suppose that $V=V^\; \{-\; 1\}\; \backslash ,$. If $x\; \backslash \; not=e\; \backslash ,$, one applies this remark to $U=G\; \backslash \; setminus\; X\; \backslash ,$. Then $V\; \backslash ,$ and $x*V\; \backslash ,$ are two open disjoined container $e\; \backslash ,$ and $x\; \backslash ,$ respectively.

• If $U\; \backslash ,$ is an open part and $A\; \backslash ,$ an unspecified part ,

$U*A\; \backslash ,$ and $A*U\; \backslash ,$ are opened, since for example $U*A=\; \backslash \; cup\_\; \{has\; \backslash \; in\; has\}\; U*a\; \backslash ,$.

Henceforth, we will omit the sign $*\; \backslash ,$.

Linear groups

An important class of groups topoloqic is formed by the sub-groups of linear group $Gl\; (N,\; K)\; \backslash ,$, with $K=\; \backslash \; mathbb\; \{R\}$ or $\backslash \; mathbb\; \{C\}$. One provides them with the topology induced by that of $End\; (K^n)\; \backslash ,$.

These examples are fundamental examples of groups of Dregs real or complex.

They in common have the property according to: there exists open containing the neutral element and not containing any sub-group noncommonplace.

P-adic topology

If ( G , +) is an abelian group, if ( G _N ) is a succession of sub-groups of G such as:

$G\; =\; G\_0\; \backslash \; supset\; G\_1\; \backslash \; supset\; G\_2\; \backslash \; supset\dots .\backslash \; supset\; G\_n\; \backslash \; supset\dots$

Then the continuation ( G _N ) induced a topology on G in which the vicinities of X are the whole X + G N .

So moreover, the intersection of the G N is reduced to {0} where 0 are the neutral element of G , the group is separate.

A topological particular case of group of this form is the group provided with the p-adic topology : If p is a natural entirety, the continuation ( G _N ) is defined by $G\_n\; =\; p^nG$. (it is pointed out that, for entire naturalness K and any element X of G , the element kx is defined by kx = X + X +… + X (where X appears K time)

Induced distance

One can define a distance on ( G , +) provided with the topology induced by ( G _N ) if the intersection of the G N is well reduced to {0}:

$d\; (X,\; there)\; =\; \backslash \; frac\; \{1\}\; \{2^k\}$ where K is the first entirety such as X - does not belong there to G K .

D ( X , there ) = 0 so for entire K , $x\; -\; there\; \backslash \; in\; G\_k$

Supplemented

If ( G , +) is a separate abelian group provided with topology determined thereafter ( G _N ), one can define in G series Cauchy

( X _N ) is of Cauchy if and only if, for any vicinity V (0) of 0, there exists an entirety N such as

for all $m\; \backslash \; geq\; n$, $x\_m-x\_n\; \backslash \; in\; V\; (0)$

On this whole of continuations S _C ( G ), one can define a relation of equivalence:

$(x\_n)\; R\; (y\_n)\; \backslash \; Leftrightarrow\; \backslash \; lim\; (x\_n\; there\; \_n)\; =0$

The unit quotient S _C ( G ) is then complete Espace.

The group G is then isomorphous with a dense sub-group of S _C ( G ).

The most important example of such a construction is that of the p-adic numbers: this construction is made starting from $\backslash \; mathbb\; \{Z\}\; \backslash ,$ and multiplication by a prime number $p\; \backslash ,$.

See too

• R. Mneimné, F. Testard, Introduction to the theory of the traditional groups of Dregs, Hermann 1986,

ISBN 2 7056 6040 2

• J. Lafontaine, Introduction to the differential varieties (CH. 4), ISBN 2 7061 0654 9

• R. Godement, Introduction to the theory of the groups of Dregs, (CH. 1 and 2), Springer

2004, IX, 305 p., Stitched, ISBN 3-540-20034-7

Internal bonds

• topological Space
• groups of Dregs
• groups profinis

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