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Math Help - is group structure preserved under homeomorphism

  1. #1
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    is group structure preserved under homeomorphism

    Hi;

    I hope you are well.

    I need your help:

    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.

    Please help me and every guidance is highly appreciated.

    Thank you in advance
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    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.
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  3. #3
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    Re: is group structure preserved under homeomorphism

    Ok, I will try to do that and see if it works. Thanks alot
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    Re: is group structure preserved under homeomorphism

    Quote Originally Posted by student2011 View Post
    Hi;

    I hope you are well.

    I need your help:

    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.

    Please help me and every guidance is highly appreciated.

    Thank you in advance
    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).
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    Re: is group structure preserved under homeomorphism

    Quote Originally Posted by student2011 View Post
    Hi;

    I hope you are well.

    I need your help:

    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.

    Please help me and every guidance is highly appreciated.

    Thank you in advance
    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.
    Last edited by SlipEternal; December 16th 2013 at 02:51 PM.
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  6. #6
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    Re: is group structure preserved under homeomorphism

    Quote Originally Posted by student2011 View Post
    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
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  7. #7
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    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).
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  8. #8
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    Re: is group structure preserved under homeomorphism

    Quote Originally Posted by student2011 View Post
    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.


    Thank you in advance
    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.
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