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Thread: Open Set

  1. #1
    Senior Member slevvio's Avatar
    Oct 2007

    Open Set

    I was wondering if someone could help with this question. I am stuck on part of the question showing that the following set is a regular surface in $\displaystyle \mathbb{R}^3$
    $\displaystyle S =\{ p + uX + vY \mid u,v \in \mathbb{R} \}, $i.e. the affine plane through a point p spanned by linearly independent vectors.

    In particular I am trying to show that the function $\displaystyle f: \mathbb{R}^2 \rightarrow S \cap \mathbb{R}^3 = S $, given by $\displaystyle f(x,y) = p + xX + yY $ is a homeomorphism. I think its clear that it is bijective and continuous but I am having difficulty showing that it's inverse is continuous, that is for some open set O $\displaystyle \subseteq \mathbb{R}^2$,

    $\displaystyle f(O) = S \cap E$ for some open set E in $\displaystyle \mathbb{R}^3$. How do I find out what E must be?

    any help would be appreciated
    Last edited by slevvio; Oct 18th 2010 at 05:16 AM.
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