Thread: is group structure preserved under homeomorphism

1. is group structure preserved under homeomorphism

Hi;

I hope you are well.

Suppose X and Y are topological homeomorphic spaces such that X is a group. Then is it true that Y is also a group?

Two spaces are said to be homeomorphic if there is a bijective continuous function between them and the inverse of the function is also continuous.

Group is algebraic structure. a set X with binary operation * is a group if (1) * is associative that is x*(y*z)=(x*y)*z for all x,y,z in X. (2) there is an element e in X called the identity element which satisfy e*x=x*e=x, for all x in X. (3) for each element x in X there is an inverse x^-1 such that x*x^-1=x^-1*x=e.

2. Re: is group structure preserved under homeomorphism

Hey student2011.

You should probably start off with the definition of a homo-morphism (or an iso-morphism for bijective spaces) and prove the group axioms one by one.

3. Re: is group structure preserved under homeomorphism

Ok, I will try to do that and see if it works. Thanks alot

4. Re: is group structure preserved under homeomorphism

Originally Posted by student2011
Hi;

I hope you are well.

Suppose X and Y are topological homeomorphic spaces such that X is a group. Then is it true that Y is also a group?

Two spaces are said to be homeomorphic if there is a bijective continuous function between them and the inverse of the function is also continuous.

Group is algebraic structure. a set X with binary operation * is a group if (1) * is associative that is x*(y*z)=(x*y)*z for all x,y,z in X. (2) there is an element e in X called the identity element which satisfy e*x=x*e=x, for all x in X. (3) for each element x in X there is an inverse x^-1 such that x*x^-1=x^-1*x=e.

Not necessarily:

Suppose X = Y = $\mathbb{R}$, with the usual topology on both, and our homeomorphism is the identity map. If the operation in Y is multiplication, and the operation in X is addition, then even though X and Y are homeomorphic as topological spaces, they aren't algebraically isomorphic, because Y isn't even a group (0 has no inverse).

5. Re: is group structure preserved under homeomorphism

Originally Posted by student2011
Hi;

I hope you are well.

Suppose X and Y are topological homeomorphic spaces such that X is a group. Then is it true that Y is also a group?

Two spaces are said to be homeomorphic if there is a bijective continuous function between them and the inverse of the function is also continuous.

Group is algebraic structure. a set X with binary operation * is a group if (1) * is associative that is x*(y*z)=(x*y)*z for all x,y,z in X. (2) there is an element e in X called the identity element which satisfy e*x=x*e=x, for all x in X. (3) for each element x in X there is an inverse x^-1 such that x*x^-1=x^-1*x=e.

You can define $Y$ to be a group. Suppose (X,*) is a group. Let $f:X \to Y$ be a bijection (it doesn't matter if it is a homeomorphism). Define a binary operator +:YxY->Y by y_1+y_2 = f(f^{-1}(y_1)* f^{-1}(y_2)). Now, it is extremely easy to check that (Y,+) is a group. Moreover, f is a homomorphism. So, your question is not well-defined. As Deveno showed, a homeomorphism is not the same as an isomorphism, so the group structures may not necessarily be the same.

Another question: Given two homeomorphic spaces, each a group with a binary operator, does there exist a homeomorphism which is also an isomorphism? That is a more complicated question. The answer is no in general. Let G and H be two non-isomorphic groups of equal cardinality. Give each the discrete topology. Then, any bijection between the two is a homeomorphism (since every function between two spaces with discrete topologies will be continuous). But, since you selected the groups to be non-isomorphic, there is no isomorphism between them.

6. Re: is group structure preserved under homeomorphism

Originally Posted by student2011
Hi;
Suppose X and Y are topological homeomorphic spaces such that X is a group. Then is it true that Y is also a group?

Two spaces are said to be homeomorphic if there is a bijective continuous function between them and the inverse of the function is also continuous.
Depends on f and *.

if f(x)=x2 and * = +
x+y=z does not imply x2+y2=z2, NO

if f(x)=x2 and * = X
xXy=z <-> x2Xy2=z2 (pos root)
1Xx=x, 12=1, 1Xx2=x2, YES

7. Re: is group structure preserved under homeomorphism

Previous reply was within context of the OP question.

If
Group: *, associativity,identity,inverse.
Homeomorphism: defined between groups and (a*b)’=a’*’b’

Then
Y is trivially a group by definition (Homeomorphism defined between groups).

8. Re: is group structure preserved under homeomorphism

Originally Posted by student2011
Hi;

Suppose X and Y are topological homeomorphic spaces such that X is a group. Then is it true that Y is also a group?

Two spaces are said to be homeomorphic if there is a bijective continuous function between them and the inverse of the function is also continuous.

Yes. You're welcome.

Homeomorphism is not a homomorphism. Definition is correct.

Let f:X →Y be a homeomorphism and let X be a group under *.

Let x’=f(x), y’=f(y), …..
Let x’*’y’=z’ where z=x*y. Then Y is a group under *’.

x’*’(y’*z’)=x*y*z.
e’*’y’=(e*y)’=y’ unit
x’*’y’=e’, e=x*y inverse

Example f(x) = x’ = x2 and * = +.
Then x’*’y’=z’ → x1/2*’y1/2=(x+y)1/2, which is a result of the definition of *’.
f, *,*’ is not necessarily a homomorphism between X and Y.