The Topology of Spacetimes: Exploring the Global Structure of Curved Manifolds

[WannabeNewton]

So there is no need for a Hausdorff pseudo-Riemannian manifold to be a metric space (ie. is it a red herring to be concerned about metric spaces in GR)?

Well manifolds are metrizable so in principle you can endow the manifold with a metric. The pseudo - Riemannian structure won't change that because the proof that topological manifolds are metrizable is, as stated, for topological manifolds which don't have any prescribed pseudo - Riemannian structure or Riemannian structure if that is what you are asking. I don't think it particularly matters in the context of GR because I've never seen a metric (as opposed to the metric tensor) ever being used in any textbook I've seen. Someone else could probably comment on that.

 

[kevinferreira]

Well manifolds are metrizable so in principle you can endow the manifold with a metric. The pseudo - Riemannian structure won't change that because the proof that topological manifolds are metrizable is, as stated, for topological manifolds which don't have any prescribed pseudo - Riemannian structure or Riemannian structure if that is what you are asking. I don't think it particularly matters in the context of GR because I've never seen a metric (as opposed to the metric tensor) ever being used in any textbook I've seen. Someone else could probably comment on that.

So we can endow spacetime itself with a metric distance, but we don't usually do it. Why not?

I mean, we do have an invariant $ds^2$, but this may be positive, negative or null. Then, what is done in physics is that we define our distances simply by
$$\int_a^bds=\int_a^b\sqrt{\pm g_{\mu\nu}dx^{\mu}dx^{\nu}}\geq 0$$
and this only makes sense if b is inside the lightcone defined by a, so that (by using the proper convention sign $\pm$) we always get a distance properly defined. This may be a simple trick by using the lightcone, but it can be supported by causality arguments that you want to include in our mode, I think. Does this seems good to you? I have some problems, as I'm thinking of the lightcone as being on the manifold, and not in the tangent space as has been argued.

 

[WannabeNewton]

Why not?

I don't know Kevin; someone else would have to answer that.

I have some problems, as I'm thinking of the lightcone as being on the manifold, and not in the tangent space as has been argued.

The terminology makes things ambiguous. Hawking and Elis clears this stuff up pretty nicely I would say. The null cone is a subset of the tangent space. The image of the null cone under the exponential map is the set of all null geodesics in M going through p which is of course a subset of M.

 

[micromass]

So we can endow spacetime itself with a metric distance, but we don't usually do it. Why not?

My guess is that the distance is just not a very useful one as it won't agree with the pseudo-Riemannian metric.

For example, take the sphere in $\mathbb{R}^3$. We can endow this with a metric as follows. Take two points on the sphere, draw a straight line through those points and measure the length of the line. So we take the distance on $\mathbb{R}^3$ and restrict it to the sphere. This defines a good distance on the sphere that agrees with the topology. However, this distance is not a very useful one as it relies on the embedding in $\mathbb{R}^3$.
What we want is a distance on the sphere that measures the length of the paths on the sphere. So we don't want a distance that comes from straight lines (which are not on the sphere), but rather a distance that comes from path (=great circles) on the sphere. This distance is a distance coming from a metric tensor and this is much more useful.

In the same way, we can endow a distance on a spacetime. But nothing tells us that this distance actually has a physical significance or that it agrees with some metric tensor.

 

[Fredrik]

So we can endow spacetime itself with a metric distance, but we don't usually do it. Why not?

I mean, we do have an invariant $ds^2$, but this may be positive, negative or null. Then, what is done in physics is that we define our distances simply by
$$\int_a^bds=\int_a^b\sqrt{\pm g_{\mu\nu}dx^{\mu}dx^{\nu}}\geq 0$$
and this only makes sense if b is inside the lightcone defined by a, so that (by using the proper convention sign $\pm$) we always get a distance properly defined. This may be a simple trick by using the lightcone, but it can be supported by causality arguments that you want to include in our mode, I think. Does this seems good to you? I have some problems, as I'm thinking of the lightcone as being on the manifold, and not in the tangent space as has been argued.

You don't integrate "from a to b", you integrate along a curve. This way you can define the length of a spacelike curve. You can also (by changing the sign under the square root in the definition) use this method to define the "length" of a timelike curve, but we call it "proper time", not "length". Since there are always infinitely many spacelike curves connecting two given spacelike separated events, and infinitely many timelike curves connecting two given timelike separated events, this doesn't immediately lead to a well-defined notion of "distance" between the two events. You could try to define the distance between two events as the length or proper time along a geodesic connecting the two events. But I think that in some spacetimes, there can be many such geodesics. And even in spacetimes where the geodesics are unique, you have to deal with events that are null separated from each other. I doubt that there's a way to define the distance between those that would give you a distance function that satisfies the requirements in the definition, like the triangle equality.

It seems to me that there is a meaningful notion of "lightcone" on the manifold as well. (Not sure what the standard terminology is though). This would be the union of all the timelike and null geodesics through the given point.

 

[micromass]

Is a Hausdorff space necessarily a metric space?

No, wbn gave the counterexample of the long line. This is a non-metric space that is Hausdorff.

Wikipedia just says thart pseudometric spaces are typically not Hausdorff, but that seems to allow that Hausdorff spaces can be neither metric nor pseudometric.

A pseudometric space is actually Hausdorff if and only if it is a metric space. So a pseudometric space that is not a metric space can never be Hausdorff.

If that is possible, then wouldn't it be possible that Hausdorff manifolds with pseudo-Riemannian metric tensors need not be metric spaces?

Well, a manifold is usually defined as a topological space that is

• Locally Euclidean
• Hausdorff
• Second countable

It can be proven (but the proof is not easy by far), that if we have these three topological conditions, then our space is metrizable. So we can always find a metric. Moreover, we can always embed our manifold in $\mathbb{R}^n$.

So even without a smooth structure or a metric tensor, we already have that our manifold is metrizable. Again: the metric of the metric space might not be physical or might not have anything to do with a metric tensor!

If we drop one of the conditions from our list, then the space is not metrizable anymore. For example, if we would define a manifold as just locally euclidean and Hausdorff, then it might not be metrizable (as the long line shows). If we define a manifold as just locally euclidean and second countable, then it might also not be metrizable (as the line with two origins shows). In fact: it might not even be pseudo-metrizable.

 

[atyy]

So even without a smooth structure or a metric tensor, we already have that our manifold is metrizable. Again: the metric of the metric space might not be physical or might not have anything to do with a metric tensor!

Would it be right to paraphrase this way: you could put a metric on a Hausdorff pseudo-Riemannian manifold (eg. via a Riemannian metric tensor or some other means not involving a metric tensor at all), but it is physically irrelevant ?

 

[micromass]

Would it be right to paraphrase this way: you could put a metric on a Hausdorff pseudo-Riemannian manifold (eg. via a Riemannian metric tensor or some other means not involving a metric tensor at all), but it is physically irrelevant ?

I think that is correct.

 

[WannabeNewton]

This would be the union of all the timelike and null geodesics through the given point.

Hi Fredrik! Correct me if I'm wrong but I'm pretty sure the "light cone" itself is just the set of all null geodesics through p and the interior consists of the time - like geodesics.

 

[zonde]

For most situations, spacetime Hausdorffness seems to be a reasonable, physical separation axiom.

No. In relativity it is not reasonable to believe that spacetime events connected with null geodesics are separable. Or let's rather say that their separability does not depend on spacetime properties but rather on distribution of content within spacetime. There is a lot of matter around one particular state of motion and that determines separability of events not spacetime properties.

 

[micromass]

No. In relativity it is not reasonable to believe that spacetime events connected with null geodesics are separable. Or let's rather say that their separability does not depend on spacetime properties but rather on distribution of content within spacetime. There is a lot of matter around one particular state of motion and that determines separability of events not spacetime properties.

Well, then I guess that Wald's textbook must be completely wrong. Do you think so? Since Wald seems to let spacetimes be Hausdorff...

 

[WannabeNewton]

No. In relativity it is not reasonable to believe that spacetime events connected with null geodesics are separable. Or let's rather say that their separability does not depend on spacetime properties but rather on distribution of content within spacetime. There is a lot of matter around one particular state of motion and that determines separability of events not spacetime properties.

I'm not sure if you are understanding what it means for a topological space to be Hausdorff. Sure two events connected by a null geodesic represent a light pulse being able to get from one to the other but what does that have to do with Hausdorff? Hausdorff simply states there exist a pair of neighborhoods, for the two (distinct) events, that are disjoint but you seem to be thinking that this implies we could not anymore connect the two events with the aforementioned null geodesic. If the null geodesic connects the two events then that is that; the Hausdorff property won't break anything.

 

[TrickyDicky]

So we can endow spacetime itself with a metric distance, but we don't usually do it. Why not?

I mean, we do have an invariant $ds^2$, but this may be positive, negative or null. Then, what is done in physics is that we define our distances simply by
$$\int_a^bds=\int_a^b\sqrt{\pm g_{\mu\nu}dx^{\mu}dx^{\nu}}\geq 0$$
and this only makes sense if b is inside the lightcone defined by a, so that (by using the proper convention sign $\pm$) we always get a distance properly defined. This may be a simple trick by using the lightcone, but it can be supported by causality arguments that you want to include in our mode, I think. Does this seems good to you? I have some problems, as I'm thinking of the lightcone as being on the manifold, and not in the tangent space as has been argued.

So let's make a distinction between the different tangent vectors (timelike,null, spacelike) in the tangent space at a point of a manifold with a pseudoRiemannian metric tensor field, that define the structure of a light cone in the tangent space, versus the different paths in a manifold that are also called timelike, spacelike or null according to what the tangent vector is at every point in the curve.
As you notice, in physics the length of the curve is always computed as if the tangent vector at every point where inside the light cone(in the limit at infinity) , regardless of what it is called, i.e. photon's null paths are never considered to have null length. There are no physical examples of spacelike paths so we can leave those out for now.
This is the logic thing to do since after all we are working with a smooth manifold that doesn't alter its topology nor its distance function(in the Riemannian manifold case) by the introduction of a pseudoRiemannian metric tensor.
The only problem I see is that this seems to be forgotten when applying GR to specific solutions of the EFE.

 

[TrickyDicky]

Would it be right to paraphrase this way: you could put a metric on a Hausdorff pseudo-Riemannian manifold (eg. via a Riemannian metric tensor or some other means not involving a metric tensor at all), but it is physically irrelevant ?

I'm not sure what you mean here, but I'd say the metric(distance function) is never physically irrelevant in GR. One of the pillars of the theory is the invariance of length across arbitrarily long distances, think of cosmological redshifts. If we didn't care about metrics (distances) in GR there would be no need for a curvature concept or unique connections.

 

[Fredrik]

Hi Fredrik! Correct me if I'm wrong but I'm pretty sure the "light cone" itself is just the set of all null geodesics through p and the interior consists of the time - like geodesics.

That seems to make more sense than what I said, and Wikipedia agrees with you. This sort of thing happens a lot when I post just before going to bed.

 

[atyy]

I'm not sure what you mean here, but I'd say the metric(distance function) is never physically irrelevant in GR. One of the pillars of the theory is the invariance of length across arbitrarily long distances, think of cosmological redshifts. If we didn't care about metrics (distances) in GR there would be no need for a curvature concept or unique connections.

In cosmology, 4D spacetime is cut into 3D spatial slices that change with time. On the 4D spacetime, there is a pseudo-Riemannian metric tensor, and no physically relevant metric space metric. On each 3D spatial slice there is a Riemannian metric tensor, which can be used to define a metric space metric.

It seems to me that there is a meaningful notion of "lightcone" on the manifold as well. (Not sure what the standard terminology is though). This would be the union of all the timelike and null geodesics through the given point.

Hi Fredrik! Correct me if I'm wrong but I'm pretty sure the "light cone" itself is just the set of all null geodesics through p and the interior consists of the time - like geodesics.

That seems to make more sense than what I said, and Wikipedia agrees with you. This sort of thing happens a lot when I post just before going to bed.

"achronal boundary"? "chronological future" or "timelike future"?

http://www.math.miami.edu/~galloway/beijing.pdf
http://en.wikipedia.org/wiki/Causal_structure

 

[stevendaryl]

... since after all we are working with a smooth manifold that doesn't alter its topology nor its distance function(in the Riemannian manifold case) by the introduction of a pseudoRiemannian metric tensor.
The only problem I see is that this seems to be forgotten when applying GR to specific solutions of the EFE.

What do you mean? In what sense does anything done in GR contradict the claim that the topology isn't altered by introducing a metric tensor?

 

[TrickyDicky]

In cosmology, 4D spacetime is cut into 3D spatial slices that change with time. On the 4D spacetime, there is a pseudo-Riemannian metric tensor,

Ok.

and no physically relevant metric space metric.

Do you really think the spacetime invariant interval between events is physically irrelevant? That's odd.

 

[TrickyDicky]

What do you mean? In what sense does anything done in GR contradict the claim that the topology isn't altered by introducing a metric tensor?

For instance, the topology of a smooth manifold (that by definition has no discontinuities) is altered by introducing singularities based precisely on the peculuarities of the pseudoriemannian metric tensor.

 

[micromass]

For instance, the topology of a smooth manifold (that by definition has no discontinuities) is altered by introducing singularities based precisely on the peculuarities of the pseudoriemannian metric tensor.

I don't get this. A manifold has a topology. Only then do we introduce a metric tensor. So the metric tensor is an extra structure.
How can an extra structure possibly change the topology of a manifold??

 

[kevinferreira]

For instance, the topology of a smooth manifold (that by definition has no discontinuities) is altered by introducing singularities based precisely on the peculuarities of the pseudoriemannian metric tensor.

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

 

[TrickyDicky]

I don't get this. A manifold has a topology. Only then do we introduce a metric tensor. So the metric tensor is an extra structure.
How can an extra structure possibly change the topology of a manifold??

That IS my point. I'm saying that it shouldn't change it.

 

[TrickyDicky]

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

Correct me if I'm wrong but singularities may be viewed as. discontinuities, at least in the wikipdia page about singularity theory they are defined as failures of the manifold structure, in which case they would affect the topology. But even in the case one decides this is not the case, it is clear that at the very least it affects the differential structure, and we wouldn't be dealing with smooth manifold in its presence.

The things you mention about GR wouldn't be affected, but all the physical assertions in GR either for cosmological or asymptotically flat cases that deal with the manifold globally would.

 

[zonde]

I'm not sure if you are understanding what it means for a topological space to be Hausdorff. Sure two events connected by a null geodesic represent a light pulse being able to get from one to the other but what does that have to do with Hausdorff? Hausdorff simply states there exist a pair of neighborhoods, for the two (distinct) events, that are disjoint but you seem to be thinking that this implies we could not anymore connect the two events with the aforementioned null geodesic. If the null geodesic connects the two events then that is that; the Hausdorff property won't break anything.

The way you say it null geodesic is just line with some idea of length along the line. Well, the idea about spacetime is that the length along null geodesic is always zero.

Hmm, maybe I would convey my view on this if I would speak about spacetime as a space whose elements are null geodesics not events. If you think about it it makes sense from physical perspective. Events have no relevance if there is no null or timelike worldline extending from it (and toward it).

 

[atyy]

Do you really think the spacetime invariant interval between events is physically irrelevant? That's odd.

Let's consider flat spacetime for simplicity. In flat spacetime the invariant interval determined by the pseudo-Riemannian metric tensor is physically relevant. It's just that this isn't the metric of a metric space. I guess there are 3 notions: pseudo-Riemannian metric, Riemannian metric, metric of a metric space. The first two are related (both are defined by their action on vectors in the tangent space at each point), the last two are related (the Riemannian metric tensor can be used to define the metric of a metric space), but the first and the last are not (the pseudo-Riemannian metric tensor cannot be used to define the metric of a metric space), and it's the first that is physically relevant in spacetime.

 

[TrickyDicky]

Let's consider flat spacetime for simplicity. In flat spacetime the invariant interval determined by the pseudo-Riemannian metric tensor is physically relevant.

Why is the spacetime interval relevant in SR and not in GR?

 

[atyy]

Why is the spacetime interval relevant in SR and not in GR?

The pseudo-Riemannian metric acts on tangent vectors at a point. To get a "distance" (which is not the distance of a metric space:) between events at different spacetime points, we have to specify a path to integrate over. In flat spacetime, there is a unique extremal path between every pair of points and we use that path to define the spacetime interval from the pseudo-Riemannian metric. I think this idea can be generalized to curved spacetime in some circumstances, but the generalization wasn't immediately obvious to me (how to choose the path?), so I restricted my discussion to flat spacetime in the earlier post to focus on the 3 different quantities (pseudo-Riemannian metric tensor, Riemannian metric tensor, metric of metric space).

 

[strangerep]

I don't get this. A manifold has a topology. [...]

This is the crucial point. A manifold is a mathematical abstraction which we use to construct models of physics. So of course, we physicists tend to assume too easily that this mathematical abstraction is homeomorphic, isomorphic, etc, to something out there in the real world.

Also, many physicists don't understand that a given set can be equipped with various inequivalent topologies -- which is strange since the distinction between strong and weak topology on a Hilbert space is something of which any self-respecting physicist ought to be at least vaguely aware. Both can be useful -- in different contexts.

What's most important for physics are the observables -- fields on the manifold. One may think of these as mappings from an abstract state space (the points on the manifold) to some more convenient linear space that's closely relatable to measurement data. Thus, they may be regarded as generalized functionals, and hence define a weak topology on the manifold state space. IMHO, such weak topologies are more important for physics because they come from physically meaningful observables.

****************************

For the benefit [or perhaps confusion?] of other readers: an ordinary "weak topology" is constructed essentially by demanding that a certain function be continuous on the underlying set. One typically does not actually construct the open sets explicitly in this weak topology, since what is important is that the function is deemed continuous -- meaning that every open set in the range of the function comes from an open set in the domain, the latter being thereby defined by the function rather than by some independent lower level construction.

 

[George Jones]

Correct me if I'm wrong but singularities may be viewed as. discontinuities, at least in the wikipdia page about singularity theory they are defined as failures of the manifold structure, in which case they would affect the topology. But even in the case one decides this is not the case, it is clear that at the very least it affects the differential structure, and we wouldn't be dealing with smooth manifold in its presence.

The things you mention about GR wouldn't be affected, but all the physical assertions in GR either for cosmological or asymptotically flat cases that deal with the manifold globally would.

I am going to give examples to try and illustrate what micromass and kevinferreira have written.

I don't get this. A manifold has a topology. Only then do we introduce a metric tensor. So the metric tensor is an extra structure.

How can an extra structure possibly change the topology of a manifold??

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

As a differentiable manifold, what is the spacetime of an open Friedmann-Lemaitre-Robertson-Walker universe? This differentiable manifold is $\mathbb{R}^4$. There is no problem with topological or manifold structure, yet this spacetime is singular.

As far as I know, there is no reasonably generic, accepted definition of "spacetime singularity". There is, however, a reasonably generic definition of "singular spacetime". A rough, sufficient condition: spacetime is singular if there is a timelike curve having bounded acceleration that ends in the past or the future after a finite amount of proper time. For example, and speaking very loosely, a spacetime is singular if a person can get in a rocket, and, after using a finite amount of fuel wristwatch time, can fall "off of spacetime" at a "singularity".

The example of an open FLRW universe shows that "singular" is due to the extra structure of a pseudo-Riemannian metric tensor field.

My guess is that the distance is just not a very useful one as it won't agree with the pseudo-Riemannian metric. ... In the same way, we can endow a distance on a spacetime. But nothing tells us that this distance actually has a physical significance or that it agrees with some metric tensor.

As $\mathbb{R}^4$, clearly, we can introduce a positive-definite distance function on open FLRW universes, but, as micromass notes, this wouln't have physical significance. The differentiable manifold together with the added structure of a particular pseudo-Riemannian metric nicely models some physics. Particular pseudo-Riemannian metric, because different pseudo-Riemannian metrics can be added to the same differentiable manifold, with very different results! For example, adding the Minkowski metric to $\mathbb{R}^4$ results in Minkowski spacetime. The same underlying differentiable manifold, yet one spacetime is singular and the other spacetime is non-singular!

I'm not sure what you mean here, but I'd say the metric(distance function) is never physically irrelevant in GR. One of the pillars of the theory is the invariance of length across arbitrarily long distances, think of cosmological redshifts. If we didn't care about metrics (distances) in GR there would be no need for a curvature concept or unique connections.

As the example of open FLRW universes demonstrates, the added pseudo-Riemannian metric is the physically significant structure that is added.

Why is the spacetime interval relevant in SR and not in GR?

The spacetime interval is very relevant physically. For example, consider an observer's worldline that joins events p and q. The worldline doesn't have to be a geodesic, as the observer could be in a rocket. How much observer wristwatch time elapses between p and q? Appropriately integrate the spacetime metric along the worldline to find out.

In cosmology, 4D spacetime is cut into 3D spatial slices that change with time. On the 4D spacetime, there is a pseudo-Riemannian metric tensor, and no physically relevant metric space metric. On each 3D spatial slice there is a Riemannian metric tensor, which can be used to define a metric space metric.

As another example, again consider FLRW universes. What is the present proper spatial distance between galaxies A and B? Appropriately integrate the spacetime metric along a path in the "now" spatial hypersurface to find out.

I think that this is a beautiful interplay between physics and mathematics.

 

[TrickyDicky]

As a differentiable manifold, what is the spacetime of an open Friedmann-Lemaitre-Robertson-Walker universe? This differentiable manifold is $\mathbb{R}^4$. There is no problem with topological or manifold structure, yet this spacetime is singular.

Ok, no problem with the coarsest topology in the topological manifold structure, but there is certainly problem with the finer topology required for the differentiable manifold (that must be Hausdorff and second-countable) that I'd like to understand how it can be compatible with singular points in a manifold. Singularities seem to be incompatible also with a global(not just local) differentiable structure.

As far as I know, there is no reasonably generic, accepted definition of "spacetime singularity". There is, however, a reasonably generic definition of "singular spacetime". A rough, sufficient condition: spacetime is singular if there is a timelike curve having bounded acceleration that ends in the past or the future after a finite amount of proper time. For example, and speaking very loosely, a spacetime is singular if a person can get in a rocket, and, after using a finite amount of fuel wristwatch time, can fall "off of spacetime" at a "singularity".

I would find really upsetting if there is no accepted definition of spacetime singularity when such an important part of GR theory deals with singularities (BHs, BBT) and so much physics literature is devoted to them.
Having said this your definition of singular spacetime might clear up something for me, is it defining something like a metric space that is not complete, that has missing points? Can this missing points be considered singularities? In that case things would start to make sense to me.

The example of an open FLRW universe shows that "singular" is due to the extra structure of a pseudo-Riemannian metric tensor field.

Sure. My confusion comes from not seeing how an structure that is supposed to act only locally can have global effects.

The spacetime interval is very relevant physically. For example, consider an observer's worldline that joins events p and q. The worldline doesn't have to be a geodesic, as the observer could be in a rocket. How much observer wristwatch time elapses between p and q? Appropriately integrate the spacetime metric along the worldline to find out.

As another example, again consider FLRW universes. What is the present proper spatial distance between galaxies A and B? Appropriately integrate the spacetime metric along a path in the "now" spatial hypersurface to find out.

I think that this is a beautiful interplay between physics and mathematics.

Agreed, that's why I found atyy's statement odd.

 

[micromass]

Ok, no problem with the coarsest topology in the topological manifold structure, but there is certainly problem with the finer topology required for the differentiable manifold (that must be Hausdorff and second-countable) that I'd like to understand how it can be compatible with singular points in a manifold.

I'm sorry, but this statement is making no sense to me.

First, there is no coarser and finer topology. The topology of a topological manifold is equal to the topology of a differentiable manifold. That is because a differentiable manifold is a topological manifold with some extra structure.

By making a topological manifold into a differentiable manifold, the topology is not changed in any way. We don't add or remove open sets.

Second, any topological manifold must already be Hausdorff and second countable by definition. The definition varies of course from author to author, but the standard condition seems to be Hausdorff and second countable.

Third, Singular points on a manifold are not a concept depending on the topology.

 

[dextercioby]

Micromass, is it true that second-countable, Hausdorff and paracompact topological manifolds are metrizable ? If so, do you know a reference for a proof ? I've been looking for this fact for about an hour or so.

Thanks!

 

[micromass]

Micromass, is it true that second-countable, Hausdorff and paracompact topological manifolds are metrizable ? If so, do you know a reference for a proof ? I've been looking for this fact for about an hour or so.

Thanks!

In fact, if M is a Hausdorff locally Euclidean space, then second countable actually implies paracompact. So there is no need to add the paracompact condition. The proof of this fact can be found in Munkres: Theorem 41.5, page 257 (note that every topological manifold is in fact regular and Lindelof).

Also, a note on terminology. A topological manifold is defined as a locally euclidean, Hausdorff and second countable space. So the term "second countable topological manifold" has some unnecessary words (as does "paracompact topological manifold")

A more general result is the Smirnov metrization theorem. This states that any paracompact, Hausdorff and locally metrizable space is actually metrizable. A proof can be found in Munkres: Theorem 42.1, page 261 This proves in particular that every topological manifold is metrizable.

 

[George Jones]

Having said this your definition of singular spacetime might clear up something for me, is it defining something like a metric space that is not complete, that has missing points? Can this missing points be considered singularities? In that case things would start to make sense to me.

Because the idea is so attractive, over the decades, there have been a number of attempts to define spacetime singularities as missing points or adjoined boundaries, but many (all?) have had various problems.

The work of Susan Scott and collaborators has possibly shown the most promise, but I know very little about this stuff. An interesting recent paper:

http://iopscience.iop.org/0264-9381/28/16/165003/

Unfortunately, at this link, the paper is behind a paywall, and I can't find it on the arXiv. I have access to it, but I haven't a chance to look at it yet.

 

[TrickyDicky]

micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds, well I'm not the only one here that has pointed out that the reference text for this stuff, the 1973 book by Ellis and Hawking cites a few examples of topological manifolds that are not Hausdorff.
Besides, I've consulted several standard texts on differential geometry and they also name the Hausdorff condition only when defining smooth/differentiable manifolds.