Definition of a circle in point set topology.

[center o bass]

The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder $\mathbb{R}\times S^1$, torus $S^1 \times S^1$ etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set $\left\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = 1\right\}$?

 

[tiny-tim]

hi center o bass! happy new year!

it is, but as a definition of a 1D space, using a 2D subset with an induced topology seems over-elaborate

would you define a (2D) mobius strip using a subset of a 3D or 4D space?

the simplest definition would be the unit interval [0,1] (with the usual topology) with 0 and 1 identified

(or $\mathbb{R}/\mathbb{N}$)

 

[center o bass]

hi center o bass! happy new year!

it is, but as a definition of a 1D space, using a 2D subset with an induced topology seems over-elaborate

would you define a (2D) mobius strip using a subset of a 3D or 4D space?

the simplest definition would be the unit interval [0,1] (with the usual topology) with 0 and 1 identified

(or $\mathbb{R}/\mathbb{N}$)

Happy new year Tim! :)
I fully agree with you, and that was part of the reason why i posed the question. And thanks for the answer! Btw: Could you prehaps elaborate a bit on how
$\mathbb{R}/\mathbb{N}$ might serve as a definition of the circle? Is it somehow intuitive?

 

[tiny-tim]

Could you prehaps elaborate a bit on how
$\mathbb{R}/\mathbb{N}$ might serve as a definition of the circle? Is it somehow intuitive?

it's infinitely many copies of [0,1], with all the endpoints identfied

 

[center o bass]

But is it homeomorphic to [0,1]? Intuitively it seems like [0,1] does not have a chance to be onto $\mathbb{R}/\mathbb{N}$.

 

[spamiam]

Don't you mean $\mathbb{R}/\mathbb{Z}$ tiny-tim? $\mathbb{N}$ is not a group, so I'm not sure I know what it means for it to act on $\mathbb{R}$...

By the way, the notation $\mathbb{R}/\mathbb{Z}$ is somewhat ambiguous: we might mean that we are identifying all points in the subspace $\mathbb{Z} \subseteq \mathbb{R}$, which would result in an infinite bouquet of circles. Instead in this case we mean that the group $\mathbb{Z}$ acts on the topological space $\mathbb{R}$ by translations: given $n \in \mathbb{Z}$, the map $\varphi_n : \mathbb{R} \to \mathbb{R}$ given by $\varphi_n(x) = x + n$ is a homeomorphism. By identifying points that lie in the same orbit (i.e., real numbers that differ by an integer), we obtain a circle!

 

[lavinia]

Don't you mean $\mathbb{R}/\mathbb{Z}$ tiny-tim? $\mathbb{N}$ is not a group, so I'm not sure I know what it means for it to act on $\mathbb{R}$...

It means the same thing as when a group acts on a space. Each element of N acts as a translation of the real line and the action satifies the same conditions

0.x = x
(n+m).x = n.(m.x)

 

[center o bass]

hi center o bass! happy new year!
it is, but as a definition of a 1D space, using a 2D subset with an induced topology seems over-elaborate
would you define a (2D) mobius strip using a subset of a 3D or 4D space?
the simplest definition would be the unit interval [0,1] (with the usual topology) with 0 and 1 identified
(or $\mathbb{R}/\mathbb{N}$)

Btw Tim: would you define the circle similarly as a topological manifold? If so, what charts would you use to define the differential structure? :)

 

[WannabeNewton]

But is it homeomorphic to [0,1]? Intuitively it seems like [0,1] does not have a chance to be onto $\mathbb{R}/\mathbb{N}$.

Sure it's homeomorphic. Have you studied quotient spaces in topology and cosets in group theory? If so the fact that the two are homeomorphic should be a routine check for you. Keep in mind that $\mathbb{R}/\mathbb{Z}$ is the coset space, not the quotient space in the topological sense. In fact the quotient space, which is obtained by collapsing $\mathbb{Z}$ to a point, would be homeomorphic to a wedge sum of countably infinite many circles (c.f. exercise 3-18 in "Introduction to Topological Manifolds"-Lee).

 

[WannabeNewton]

Btw Tim: would you define the circle similarly as a topological manifold? If so, what charts would you use to define the differential structure? :)

A smooth atlas (differentiable structure) is not the same thing as a topological atlas because the former requires smooth compatibility of charts so keep that in mind. The example of $S^n$ is an extremely standard one that you will find in literally every book on topological and smooth manifolds. See, for example, p.4 of "Differential Topology"-Guillemin and Pollack.

 

[center o bass]

A smooth atlas (differentiable structure) is not the same thing as a topological atlas because the former requires smooth compatibility of charts so keep that in mind. The example of $S^n$ is an extremely standard one that you will find in literally every book on topological and smooth manifolds. See, for example, p.4 of "Differential Topology"-Guillemin and Pollack.

Yes and in all books that I've seen they define it as a set of points in R^(n+1). But like in the case of a circle in topology, isn't that superfluous? Could one not use the unit interval with identification as above and in addition specify an atlas? (I.e. a differential structure.)

 

[WannabeNewton]

Why is that superfluous? That's the most natural definition of the unit circle $S^1$ i.e. as the set of all points a unit distance from an arbitrarily chosen origin in the affine space $\mathbb{R}^2$. The topological space that you get by considering $I/\sim$, where $\sim$ is the equivalence relation $0\sim 1$, is homeomorphic to $S^1$ (as defined in the previous sentence) so the two spaces are the exact same as far as topological structure is concerned. You aren't going to lose anything in showing $S^1$ is a topological manifold using charts in $\mathbb{R}^2$; it's the easiest way to do it and since $S^1$ is homeomorphic to $I/\sim$ there's nothing wrong in making life easier.

 

[economicsnerd]

Silly question:
If Y\subseteq X (say for topological spaces with no other structure), is it standard notation to write X/Y for the space X/\sim, where \sim:=\{(x,y)\in X^2:\enspace x=y \text{ or } x,y\in Y\}? I'm not used to it, so the unit circle is the only guess I'd've had for \mathbb R/\mathbb Z.

 

[jgens]

If Y\subseteq X (say for topological spaces with no other structure), is it standard notation to write X/Y for the space X/\sim, where \sim:=\{(x,y)\in X^2:\enspace x=y \text{ or } x,y\in Y\}?

Yes.

 

[lavinia]

A circle in geometry is the set of points in a plane that are equidistant to a single point. Circles can be centered anywhere in the plane and can have any radius.

The set of points of distance 1 to the origin is one example of a circle.

If one identifies the end points of the unit interval, one obtains a topological space. It is not a circle. Rather it is homeomorphic to a circle, in fact to any circle at all.

A one dimensional closed connected manifold without boundary is another topological space. It is not a circle either. One can prove that any such manifold is homeomorphic to a circle and also to the unit interval with its end point identified.

R/Z usually denotes the quotient group of the real numbers under addition by the integers. Since R is s topological space and since the integers act properly discontinuously on it, R/Z is also a topological space and it is homeomorphic to a circle. If one takes the homomorphism x -> exp(ix) the one gets a covering of the unit complex numbers by the real line and it is easy to use this to show that R/Z is homeomorphic to the unit complex numbers which in turn are just the unit circle in the plane.

In Topology, one often thinks not of specific spaces but of all spaces that are homeomorphic to a specific space. This leads to language that calls spaces such as R/Z or the interval with is end points identified as circles. A topologist would also call an ellipse a circle.

 

[center o bass]

A circle in geometry is the set of points in a plane that are equidistant to a single point. Circles can be centered anywhere in the plane and can have any radius.

The set of points of distance 1 to the origin is one example of a circle.

If one identifies the end points of the unit interval, one obtains a topological space. It is not a circle. Rather it is homeomorphic to a circle, in fact to any circle at all.

A one dimensional closed connected manifold without boundary is another topological space. It is not a circle either. One can prove that any such manifold is homeomorphic to a circle and also to the unit interval with its end point identified.

R/Z usually denotes the quotient group of the real numbers under addition by the integers. Since R is s topological space and since the integers act properly discontinuously on it, R/Z is also a topological space and it is homeomorphic to a circle. If one takes the homomorphism x -> exp(ix) the one gets a covering of the unit complex numbers by the real line and it is easy to use this to show that R/Z is homeomorphic to the unit complex numbers which in turn are just the unit circle in the plane.

In Topology, one often thinks not of specific spaces but of all spaces that are homeomorphic to a specific space. This leads to language that calls spaces such as R/Z or the interval with is end points identified as circles. A topologist would also call an ellipse a circle.

A circle is equivalent with an ellipse as a manifold; I.e. a differential topologists would see these as different. It's only when curvature is taken into account (when one specifies a metric tensor), that the circle and the ellipse is equivalent.

 

[jgens]

A circle is equivalent with an ellipse as a manifold; I.e. a differential topologists would see these as different. It's only when curvature is taken into account (when one specifies a metric tensor), that the circle and the ellipse is equivalent.

You have this backwards. From the perspective of differential topology a circle and an ellipse are equivalent. From the perspective of differential geometry the two are different since a circle has constant curvature while an ellipse does not.

 

[center o bass]

Why is that superfluous? That's the most natural definition of the unit circle $S^1$ i.e. as the set of all points a unit distance from an arbitrarily chosen origin in the affine space $\mathbb{R}^2$. The topological space that you get by considering $I/\sim$, where $\sim$ is the equivalence relation $0\sim 1$, is homeomorphic to $S^1$ (as defined in the previous sentence) so the two spaces are the exact same as far as topological structure is concerned. You aren't going to lose anything in showing $S^1$ is a topological manifold using charts in $\mathbb{R}^2$; it's the easiest way to do it and since $S^1$ is homeomorphic to $I/\sim$ there's nothing wrong in making life easier.

I can think of some reasons: Since a manifold is a topological space with a differential structure, it has no apriori notion of distance, only the open sets and implied neighborhoods in the topology. Thus 'unit distance' is not a well defined notion as in the case with the topological circle. There is neither a notion of curvature, so the circle is equivalent to any ellipse as a manifold. Lastly, the circle is a one-dimensional manifold, and one should be able to make an 'intrinsic' definition without reference to a higher dimensional space.

I do however agree that there is nothing wrong with making life easier, but I worry that one might miss the above points by using the unit circle definition. (As I did.) I'm seeking a definition which put emphasis on the circle as a general manifold. (Not as a Riemannian one.)

I guess I could rephrase my question as: Can one define the circle, as a manifold, as the unit interval with points identified? In that case what charts should one use to specify it's differential structure?

 

[jgens]

I can think of some reasons: Since a manifold is a topological space with a differential structure, it has no apriori notion of distance, only the open sets and implied neighborhoods in the topology. Thus 'unit distance' is not a well defined notion as in the case with the topological circle. There is neither a notion of curvature, so the circle is equivalent to any ellipse as a manifold. Lastly, the circle is a one-dimensional manifold, and one should be able to make an 'intrinsic' definition without reference to a higher dimensional space.

This is just like complaining that the standard definition of Rⁿ comes with too much extra structure. Sure our natural choice of Sⁿ comes with some additional structure but this is hardly a bad thing.

I'm seeking a definition which put emphasis on the circle as a general manifold.

The standard definition as unit vectors does this. Whether the emphasis is as a differentiable or Riemannian manifold depends on the application and not the definition.

I guess I could rephrase my question as: Can one define the circle, as a manifold, as the unit interval with points identified? In that case what charts should one use to specify it's differential structure?

Yes this can be done. The charts are pretty easy to find.

 

[center o bass]

This is just like complaining that the standard definition of Rⁿ comes with too much extra structure. Sure our natural choice of Sⁿ comes with some additional structure but this is hardly a bad thing.

It might be a bad thing if there is a risk that one might think there is something special about the unit circle from the point of view of general manifolds. I think that a definition, which does not have any additional structure, has the benefit that it emphasizes what properties a circle have, and do not have, as a general manifold.

If you look back at how this thread started with my question and TinyTim's answer we came to the agreement that a definition of a circle as the points in R^2 a unit distance away from the origin is superfluous from the point of view of topology. If one agrees with that, I do not think it is a far stretch to also agree that the same definition from the point of view of manifolds is also superfluous.

I do not think however that this definition, from the point of view of differential geometry (Riemannian manifolds), is superfluous, since then the 'canonical metric tensor' of the surface is the induced metric tensor the circle inherits from being a subspace of R^n with it's Euclidean metric tensor.

 

[jgens]

It might be a bad thing if there is a risk that one might think there is something special about the unit circle from the point of view of general manifolds. I think that a definition, which does not have any additional structure, has the benefit that it emphasizes what properties a circle have, and do not have, as a general manifold.

If you look back at how this thread started with my question and TinyTim's answer we came to the agreement that a definition of a circle as the points in R^2 a unit distance away from the origin is superfluous from the point of view of topology. If one agrees with that, I do not think it is a far stretch to also agree that the same definition from the point of view of manifolds is also superfluous.

The standard definition of Rⁿ suffers from the same "defects" from your point of view. So why single out the circle?

 

[center o bass]

The standard definition of Rⁿ suffers from the same "defects" from your point of view. So why single out the circle?

I did not mean to single out the circle. What is the 'standard definition of R^n' you refer to? The space of all n-tuples $(a_1, ... a_n)$ where the $a_i \in \mathbb{R}$ with it's usual topology? This definition does not mention distance, differential structure, nor a higher dimensional space. Thus it does not suffer the same defects.

 

[jgens]

This definition does not mention distance, differential structure, nor a higher dimensional space.

Its topology comes from a metric structure on Rⁿ so the notion of distance is automatic. The definition of the unit sphere carries no intrinsic differential structure either so its not like the sphere and Euclidean space differ from this standpoint. The fact that the n-sphere does not embed in a Euclidean space of dimension n is the only true difference mentioned above.

 

[center o bass]

Its topology comes from a metric structure on Rⁿ so the notion of distance is automatic. The definition of the unit sphere carries no intrinsic differential structure either so its not like the sphere and Euclidean space differ from this standpoint. The fact that the n-sphere does not embed in a Euclidean space of dimension n is the only true difference mentioned above.

No it's not automatic because there is infinitely different metrics which induces exactly the same topology.

 

[jgens]

No it's not automatic because there is infinitely different metrics which induces exactly the same topology.

So if we simply emphasized that the sphere can be constructed and topologized with infinitely many norms/metrics that would solve all of your problems?

 

[center o bass]

So if we simply emphasized that the sphere can be constructed and topologized with infinitely many norms/metrics that would solve all of your problems?

No. The whole point of topology, as I understand it, is to generalize, so we don't need to mention any distance notion at all. All results of topology apply no matter what metric space we talk about and even to more general spaces. (Called topological spaces.) In defining the sphere as a topological space, we thus only need to specify it's topological properties. Thus I think it's good to define a sphere (circle, etc..) in terms that lay emphasis on these properties.

 

[jgens]

Thus I think it's good to define a sphere (circle, etc..) in terms that lay emphasis on these properties.

My point is that this is largely futile. Most of the interesting examples from topology are induced by other structures. For example defining the topology on Rⁿ without metric structures (or at the very least the ordering of R) is extremely difficult and not especially illuminating. Insisting that you define the sphere without ever referencing these structures is similarly unnatural. Yes the definition of Sⁿ = Dⁿ/∂Dⁿ hides the metric structure on the disc, but the fact is that the topology on Dⁿ (and in fact its definition) depends on a metric. So the point of all this is that if your goal is to emphasize the topology by eliminating any need for the natural metric structures, then you have your work cut out for you. It's much better to just accept these things and use them when convenient.

 

[center o bass]

It's much better to just accept these things and use them when convenient.

Okay so you just disagree. That is okay.

 

[jgens]

I think you miss the point. If additional structures are going to cause you grief, then can you topologize Rⁿ without appealing to a metric or order structure?

 

[center o bass]

I think you miss the point. If additional structures are going to cause you grief, then can you topologize Rⁿ without appealing to a metric or order structure?

And I think you miss the point. There is no grief, and yes, in principle that is possible, by specifying the set's of it's topology.

 

[jgens]

There is no grief

Then why start a thread about avoiding metric structures on the sphere?!

and yes, in principle that is possible, by specifying the set's of it's topology.

Now you have my interest. Would you mind specifying these sets without referencing a metric on Rⁿ or the ordering on R?

 

[center o bass]

Then why start a thread about avoiding metric structures on the sphere?!

I did not! If you had looked back, which I asked you to earlier, you would see that the thread was started with a question on how one might define a circle in topology. TinyTim answered in terms which did not involve the unit circle in R^2 and I felt it a natural progression to ask how one could do the same for a circle as a manifold. This was the point. You came in by arresting me on a terminological error.

Now you have my interest. Would you mind specifying these sets without referencing a metric on Rⁿ or the ordering on R?

Yes here is the pair $(\mathbb{R}^n, \mathcal{T})$. There is often a difference between principle and practice.

 

[jgens]

I did not! If you had looked back, which I asked you to earlier, you would see that the thread was started with a question on how one might define a circle in topology.

I still contend that defining S¹ = R/Z or S¹ = I/∂I suffers from the same defects as the unit circle definition. But at this point in time I am more interested in your later claim...

Yes here is the pair $(\mathbb{R}^n, \mathcal{T})$. There is often a difference between principle and practice.

But what are the sets in T? You claimed you could tell me what sets belong in T without reference to a metric on Rⁿ or the ordering on R. I am still waiting for this.

 

[center o bass]

I still contend that defining S¹ = R/Z or S¹ = I/∂I suffers from the same defects as the unit circle definition. But at this point in time I am more interested in your later claim...

I'm not, because that was not the point of the thread.

But what are the sets in T? You claimed you could tell me what sets belong in T without reference to a metric on Rⁿ or the ordering on R. I am still waiting for this.

No. I claim that in principle we can topologize any space $X$ by providing a pair $(X, \mathcal{T})$. How one might construct $\mathcal{T}$ in practice is another question.

 

[jgens]

How one might construct $\mathcal{T}$ in practice is another question.

With this in mind hopefully you can understand why I contend that virtually any definition of S¹ is going to require some additional structure in its definition. You can hide or obscure that structure a small amount but it is still there. In any case, just a friendly FYI if you want a definition of S¹ with a minimal amount of structure, then the R/Z definition you have been championing is not the best example. That definition induces a group structure on the circle which is something not all manifolds have.

Edit: Basically I am saying that any definition of S¹ you choose is just as good as another from a point-set topological standpoint. They all have their uses in point-set topology for different problems. To single one definition out as problematic because it contains extra structure is silly because virtually any way of topologizing the circle gives you a bit more information than its topology alone.

 

[lavinia]

I guess I could rephrase my question as: Can one define the circle, as a manifold, as the unit interval with points identified? In that case what charts should one use to specify it's differential structure?

No. The unit interval with its points identified does not define a manifold because it does not provide coordinate charts. You need to show that it can be given the structure of a manifold

 

[lavinia]

BTW: The unit interval with end points identified is an example of a structure called a CW complex. A homeomorphism from this space to a circle shows that a circle is a CW complex.

 

[center o bass]

No. The unit interval with its points identified does not define a manifold because it does not provide coordinate charts. You need to show that it can be given the structure of a manifold

Alright. But you can provide charts for it, can't you? I.e. one can give it a differential structure. How would that be done?

 

[jgens]

Let U₁ = (0,1) and U₂ = [0,1/2)∪(1/2,1]. Let φ₁ = id and φ₂|[0,1/2) = id and φ₂|(1/2,1] = 1-id. Under the 0~1 identification these descend to a pair of charts on the circle.