Related to a more mathematically inclined question, I'd like to ask the following question:
I wonder if they may have used it in a pre-topological context:
They knew/presupposed that for each length $r$ (arbitrary large or small) and each point $P$, a circle with center $P$ and radius $r$ can be drawn and is "contained in space". This can be thought of as "every circle is collapsable to a point which itself is contained in space". (In modern terms every circle is continuously collapsable to a point.)
Is there any evidence that they were aware of this characteristic feature of (Euclidean) space? And did they actually believe to have a proof of this?
Or was it too abstract for them to think about, or too "natural" to even mention it? (They probably didn't think about toroidal spaces.)
To sum it up:
Were the ancient Greeks aware of the topology of (Euclidean) space?