Q:

Convex sets in $\mathbb R$

Upto homeomorphism, how many convex sets are in $\mathbb R?$
My point of view: 
In $\mathbb R,$ there are four types of convex sets. $i.e.,(a,b),[a,b],[a,b),(a,b],$ where $ a,b \in \mathbb R.$
Since continuous image of compact set is compact, $(a,b)$ and $[a,b]$ are not homeomorphic. So  $\mathbb R$ has at least two   convex sets upto homeomorphism.
How to proceed further?
Thanks in Advance

A:

The only compact one is $[a,b]$. Note that if you remove any point from $(a,b)$ the result is not connected. This is not true of $(a,b]$, so $(a,b)$ and $(a,b]$ are not homeomophic. It's clear that $(a,b]$ and $[a,b)$ are homeomorphic.
So: three.

