Topology question for maths buffs?

Question

It is true to say that no point within a circle can touch the boundary (the closest point to the boundary is infinitessimally small - any point which appears to touch the boundary IS the boundary and is not within the circle).

Is it similarly true to say that no point exists at the centre of the circle? My intuition tells me 'no', but I'm looking for an expert's opinion.

Answer 1

My topology is pretty shaky, but maybe I can shed some light.

You are using "circle" in the sense of open disc (or 2-ball). Your first paragraph amounts to saying that the interior of an open set is disjoint from its boundary.

Now, the most common way to define a disc involves giving the center as part of the set, so asking whether the center exists in the set is trivial. But you can imagine ways of defining a disc which do not mention the center directly. If such a definition is to be equivalent to the usual definition of a disc, then the center must be in the set described (by extensionality) but let's ignore that.

If you define the center as a point equidistant from the boundary, then you can show that that point is in the interior by considering the neighborhoods around it. Showing that there exist neighborhoods around the center and in the disc would need to invoke the particulars of your alternative definition of disc, and I cannot think of a good one offhand.

I think you could also show this handily using limits of sequences, where the center would be a limit in the disc, while boundary points would not.

Answer 2

It all depends on definitions, a common definition of a circle is:
Given a point M and a number r>0, then
1)
You can define a circle C is the set of points P with distance(MP)=r.
Now a circle is just a "boundary".
2)
If you define a circle as
C is the set of points P with distance(MP)<=r.
(<= the distance is less than or equal to r), then a circle is a "disk" including the "boundary" (and consequently points on the boundary belong to the circle).
3)
if you define:
C is the set of points P with distance(MP) then the boundary doesn't belong to the circle.

The midpoint M has distance 0 (zero) to itself, so
with definitions 2) and 3) it belongs to every circle around it,
but of course it belongs not to the circle if you use def. 1)

In our daily use of language, there is no problem:
If we draw a circle with radius 5, we draw the boundary.
If we colour that circle red, we colour the disk,
(and when I was a child, my mother said "stay within the lines", so she used definition 3)

Answer 3

Of course there is a point at the center of a circle. Pick 3 points on the circle. connect the 3 points forming a triangle. Construct the perpendicular bisector of 2 sides of the triangle. The 2 perpendicular bisectors must intersect at a point which is equidistant from the three original points. that's the center of the circle.