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Math Help - What is an open mapping?

  1. #1
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    What is an open mapping?

    The definition of an open mapping is that every open set is mapped to an open set.
    But I can't have an intuitive image of open mapping.

    For the case of continuous mappings, I can say that a continuous mapping maps "points near to each other" to "points near to each other".

    If an open mapping is a bijection, then its inverse mapping is continuous, so I can say an open mapping maps "points NOT near to each other" to "points NOT near to each other".
    This is consistent with the following facts:
    "The more open sets the domain has, the more continuous mappings there are."
    "The fewer open sets the domain has, the more open mappings there are."

    But if an open mapping isn't bijective, like  f(x) = \exp(2 \pi i x) , I can't interpret it in the above sense.

    Would you please tell me (in a simple intuitive way) what an open mapping is?
    Thank you.
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  2. #2
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    The way I think about it is in terms of interior points. If A is an open subset of the domain, then every point x in A is "insulated" by a small neighborhood of other points in A. If f is an open mapping, then the same is true for f(x): it is insulated by a small neighborhood of points in f(A).
    For example, f(x)=x^2 isn't an open mapping of R -> R: A = (-1, 1) is an open set, but f(A) is not, since f(0) becomes stranded on the boundary of f(A) = [0, 1).
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  3. #3
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    You could say an open map preserves inner points.
    That is:

    f is open iff
    for all x and for all sets M
    if x is an inner point of M, then f(x) is also an inner point of f(M).

    Maybe that helps.
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  4. #4
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    I see.
    Open mappings preserve interior points.

    Thank you for both of replies.
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