Nesting of 2-Spheres & 2-Tori in Topological Spaces

[Wendel]

Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?

 

[Orodruin]

Do you mean that one sphere should be inside the other like in an onion (although not touching)? What do you think?

 

[Wendel]

Perhaps you could have A, B, C concentric in ℝ³ , describe another 2-sphere D outside of C, then identify all points in D with the point which is the center of sphere A?

 

[Orodruin]

I am sorry, but it is completely unclear to me what it is that you are trying to do. First you talked about three spheres and now you are adding another? What is your purpose?

 

[fresh_42]

Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?

You can certainly have those three nested spheres and identify the points of the outer sphere with those of the inner which makes them two again. As long as we don't have to bother embeddings, there is few which cannot carry a topology.

 

[Wendel]

Or rather take the ball
$$B = \{(x,y,z) \in ℝ^3 : x^2 +y^2 +z^2 ≤R^2 \} ⊆ ℝ^3$$
with R>0.
Now choose three smaller numbers $0 ≤ r_1 ≤ r_2 ≤ r_3 < R$ and consider the spheres
$$S^2_i = \{(x,y,z) \in ℝ^3 : x^2 + y^2 + z^2 = r^2_i\}, i \in \{1,2,3\}$$
Now Identify the boundary of $B$ with its center $(0,0,0)$. I desire to create a space where the three spheres are nested and not touching, but in such a way that $S^2_3$ appears to be inside $S^2_1$ again, analogous to how three parallels on a torus would appear to a bug on the surface, withe first "inside" the second, the second inside the third, and the third again inside the first.

 

[Orodruin]

Now Identify the boundary of $B$ with its center $(0,0,0)$.

This is probably not what you want to do. Your outer sphere will be a single point since they are all identified with the origin. You might want to identify the outer sphere with the inner sphere. This will create a topological space similar to how, in two dimensions, taking an annulus and identifying its inner and outer boundary with each other will give you a torus. However, you should be careful with concepts such as "inside" and "outside" without giving them proper definition.

 

[Wendel]

I think I understand now. Thank you fresh and Orodruin.

 

[WWGD]

Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?

What do you mean by nested , do you mean contained? In $\mathbb R^n$ for higher n , it is easier for things like these to happen, if possible. But, yes, who knows what may happen if you embed objects in spaces like Klein bottles, Projective spaces, etc. You may want to look into intersection theory ( in even dimensions in Topology; I don't know much about the Intersection theory in Algebraic Geometry, ask @mathwonk for that) in higher dimensions, to inform you if and how embedded ( homological classes of ) objects in manifolds may intersect or avoid each other. EDIT: There are invariants like Chern classes ( defined on Homology) which describe restrictions on what can happen in your manifold. Outside of the world of manifolds things are rougher. Maybe you can look into Geometric Measure Theory for non-manifolds..