TITLE: Closed Convex Subsets of $\Bbb R^2$; Find them all!
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: I think that the other four examples are:

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

REPLY [2 votes]: Hint: the other four are unbounded.