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I'm sorry if I put this in the wrong area, the author has a strange habit of going on tangents. This is Question 66 in chapter 2 of Pugh's Real Analysis.

Find all the closed and convex subsets of $\Bbb R^2$ up to homeomorphism. There are nine.

I suspect I have 5: $$\varnothing, \Bbb R^2, \{a\}, [a, b]$$ and the inclusive unit ball. Can anyone help me with the rest?

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Also see: Closed Convex sets of $ \mathbb{R}^2$. – Olivier Cailloux Aug 30 '15 at 19:24
up vote 4 down vote accepted

I think that the other four examples are:

  1. The closed upper-half plane $ \mathbb{R} \times [0,\infty) $.

  2. The line $ \mathbb{R} \times \{ 0 \} $.

  3. The line $ [0,\infty) \times \{ 0 \} $.

  4. The infinite strip $ [-1,1] \times \mathbb{R} $.

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Hint: the other four are unbounded.

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Thanks, I forgot that [a, b] is not homeomorphic to (-infinity, infinity). – Pax Kivimae Dec 21 '12 at 6:43

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