Related to a more mathematically inclined question, I'd like to ask the following question:

The ancient Greeks made use of infinite arguments and processes (limits), e.g. in the method of exhaustion and in proofs by infinite descent.

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?

  • $\begingroup$ You have to be careful with "topology" pre-19th century, one no more uses topology in collapsing circle to a point than uses analysis in intersecting lines and circles. Greeks certainly did not think relativistically about "spaces" despite studying spherical geometry and even curves on a torus, see Perseus. $\endgroup$ – Conifold Nov 6 '18 at 18:57
  • $\begingroup$ @Conifold: What exactly do you mean with "one no more uses..."? $\endgroup$ – Hans-Peter Stricker Nov 6 '18 at 19:14
  • $\begingroup$ @Conifold: Thanks a lot for your hint to Perseus whom I was not aware of! This gives me a good reading. $\endgroup$ – Hans-Peter Stricker Nov 6 '18 at 19:20
  • $\begingroup$ In my opinion studying spherical geometry and curves on a torus is thinking relativistically about "spaces". What else would qualify? $\endgroup$ – Hans-Peter Stricker Nov 6 '18 at 19:23
  • $\begingroup$ You are modernizing too much, if Greeks thought of spheres as alternative planes they would not have been obsessed with proving the parallel postulate. The relativistic view where "space", "line", "point" are empty shells constrained by axioms only was Hilbert's invention that was an ideological coup, from Greeks up to 19th century they had absolute meanings. The same goes for topology and analysis, just because we can interpret things in those terms today does not mean that anyone was using them "implicitly" before they were introduced. $\endgroup$ – Conifold Nov 6 '18 at 20:32

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