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

Question

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?

Also see: Closed Convex sets of $ \mathbb{R}^2$. – Olivier Cailloux Aug 30 '15 at 19:24 — Olivier Cailloux, Aug 30 '15 at 19:24

Haskell Curry · Accepted Answer · 2012-12-21T06:48:20.793

3

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} $.

edited Dec 21 '12 at 06:48

answered Dec 21 '12 at 06:36

Haskell Curry

19,524

score 2 · Answer 2 · answered Dec 21 '12 at 06:36

2

Hint: the other four are unbounded.

answered Dec 21 '12 at 06:36

Robert Israel

448,999

1

Thanks, I forgot that [a, b] is not homeomorphic to (-infinity, infinity). – Pax Dec 21 '12 at 06:43

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

2 Answers2