I am trying to proof that there is no homeomorphism between the closed unit disk and the real plane but i can't.Please help me.
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2What sort of topological invariants do you know? – Randall Aug 28 '17 at 11:46
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1Hint: Closed unit disk is closed and bounded. And what does this mean in real spaces? – user160738 Aug 28 '17 at 11:51
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I have seen these posts but i have no idea of homotopy theory. I have read the Munkre 's General topology. – user391120 Aug 28 '17 at 11:53
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Since you are in $\mathbb{R}^2$, a subset $C \subseteq \mathbb{R}^2$ is compact if and only if it is closed and bounded. Now, compactness is a topological invariant, hence if the closed disk were homeomorphic to the plane, you would say that the plane is compact, absurd.
TheWanderer
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Is there any ideas why any open subset of the closed unit disk could not be homeomorphic to real plane? – user391120 Aug 28 '17 at 12:00
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1@user391120 Any connected open subset of the closed unit disk is homeomorphic to the real plane. Take for instance the interior of the closed unit disk. – SvanN Aug 28 '17 at 12:02
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@ S. van Nigtevecht:Even if it consists of a part of the boundary of the closed unit circle? – user391120 Aug 28 '17 at 12:18
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@user391120: It's not true that any open subset of the closed unit disc is homeomorphic to the plane. The open disc minus the origin is a counterexample. If you wish to explore this issue further, you ought to post a separate question. – Lee Mosher Aug 28 '17 at 14:04