- #36
mathwonk
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i think you are assuming that "homeo" means homeo to an open set, but all you know is homeo to some set with the induced topology. i.e. your homeo only maps an open set to the intersection of an open set in R^2 with the image set. that is what an open set of the image means. this is very unintuitive i agree, but i think it is unavoidable.
in fact it is not true that a homeo maps interior to interior. take a homeo from an open disc in the plane to an open disc in R^3, which has no interior. it is still a homeo onto its image but the image has empty interior in R^3. you have to prove that cannot happen when the image is in R^2. it is not at all obvious. well it depends what you mean by obvious. it seems obvious but it is not easy to prove.
in fact it is not true that a homeo maps interior to interior. take a homeo from an open disc in the plane to an open disc in R^3, which has no interior. it is still a homeo onto its image but the image has empty interior in R^3. you have to prove that cannot happen when the image is in R^2. it is not at all obvious. well it depends what you mean by obvious. it seems obvious but it is not easy to prove.