# Toplogical group

1. Jul 2, 2011

### NeroKid

1. The problem statement, all variables and given/known data
Let H be a subspace of G. show that if H is also a subgroup of G then the closure of H is a topological group
G is a topological group

2. Relevant equations

3. The attempt at a solution
let closure of H is a subspace of G then the map of the operation of G restricted to cl(H) is continuous
I have the A2 axiom satisfied for cl(H) but I cant prove the the A1 + A3 +A4

2. Jul 2, 2011

### Klockan3

A1-4 doesn't tell us anything, please write them down. It is hard to help you if we don't know which definition of a topological group you use.

3. Jul 2, 2011

### NeroKid

A1: for all x y in G xRy is in G
A2 : for all x,y ,z in G xR(yRz) = (xRy)Rz
A3 : for all x in G there exists a 0 in G such that xR0=0Rx=x
A4: xRx^-1 = 0 = x^-1 R x

4. Jul 2, 2011

### micromass

Staff Emeritus
Hi NeroKid!

You know that

$$f:G\times G\rightarrow G:(x,y)\rightarrow x*y$$

is continuous. And since H is a subgroup, you know that

$$f(H\times H)\subseteq H$$

Now, what can you say about

$$f(cl(H)\times cl(H))$$

5. Jul 2, 2011

### NeroKid

tks i get it know ;))