# Closed Convex Subsets of $\Bbb R^2$; Find them all!

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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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). –  Chanler Dec 21 '12 at 6:43