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Proofs for the Open Mapping Theorem that I've seen use the Baire Category Theorem to produce an open set in $Y$ that is in the image of some scaled ball from $X$. I think I'm missing a technical point though -- why can't we take for granted that an open subset is available in the image?

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    $\begingroup$ This seems like an absurd question. All things in mathematics must be justified. You can't just take things for granted because you feel like it. $\endgroup$ – mathworker21 Apr 8 '18 at 23:26
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    $\begingroup$ I assume you mean 'why isn't it obvious?' not 'why does it need to be justified?' but you haven't explained why you think it's obvious. $\endgroup$ – spaceisdarkgreen Apr 8 '18 at 23:36
  • $\begingroup$ okay, as I tried to explain why I thought it was obvious, I realized there was something to check. The image of a ball will be non-empty in $Y$, but ensuring that the image has an interior is non-trivial. A function from the reals into the rational numbers is a function whose image has an empty interior. $\endgroup$ – yoshi Apr 8 '18 at 23:49
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    $\begingroup$ @yoshi yep, sounds like you get it now. $\endgroup$ – spaceisdarkgreen Apr 8 '18 at 23:55

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