What is a topology intuitively?

Main Question or Discussion Point

I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the additional structure of an open set of open subsets of these points), I'm really struggling to understand intuitively why this is needed, particularly in relation to manifolds used in physical applications such as GR?! I've heard explanations such as, a topology provides a primitive notion of geometry to a set without introducing additional structure such as a metric etc, and that it in some sense quantifies the "nearness" of points to one another (in the set) without having a notion of distance, in terms of neighbourhoods of points in the set, but these seem a little unsatisfactory to me.
Does one require a topology to be defined on a manifold such that the is a well-defined why to "stitch together" local coordinate patches to construct a global structure that is the manifold? Is it also required to define a notion of continuity on the manifold?

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Are you familiar with metric spaces? To understand topology, you first must understand metric spaces very well.

Are you familiar with metric spaces? To understand topology, you first must understand metric spaces very well.
Yes, I have some understanding (it's basically a set with a metric $d: (x,y)\mapsto d(x,y)$ defined on it, i.e. it is a set with a well-defined notion of distance between all elements in the set, isn't it)?!

Yes, I have some understanding (it's basically a set with a metric $d: (x,y)\mapsto d(x,y)$ on it, isn't it)?!
Yes, it is. But you need a rather full understanding of this before you can do topology. And to really understand metric spaces, you should probably understand analysis on $\mathbb{R}^n$ first. Topology will be a generalization of all of this, so it makes no sense to do topology without a rather complete understanding of metric spaces.

Once you understand metric spaces fully, you can do topology. Many notions from topology are rather obvious generalization from metric spaces. So you can see topology as a way to do analysis in metric spaces without having the bothersome metric to screw things up. In GR, every space is a metric space anyway, but the metric is often too annoying and too unphysical to use.

Yes, it is. But you need a rather full understanding of this before you can do topology. And to really understand metric spaces, you should probably understand analysis on $\mathbb{R}^n$ first. Topology will be a generalization of all of this, so it makes no sense to do topology without a rather complete understanding of metric spaces.

Once you understand metric spaces fully, you can do topology. Many notions from topology are rather obvious generalization from metric spaces. So you can see topology as a way to do analysis in metric spaces without having the bothersome metric to screw things up. In GR, every space is a metric space anyway, but the metric is often too annoying and too unphysical to use.
Fair enough. I would say I have a reasonable understanding of metric spaces, in terms of learning about open balls, open/closed sets, boundaries, homeomorphisms, etc. but I suspect this may not be quite deep enough?!

What confuses me, is that whenever I read a text on GR, the author always starts by introducing spacetime as a 4D topological manifold. There is a mention of it requiring to have a topology defined on it with the Hausdorff property (i.e. two given distinct points in the set can be separated into disjoint neighbourhoods) and being second countable (i.e. the base of the topology is countable - I have to admit, I'm not particularly familiar with this), but I'm unsure as to the significance of this since one only considers metric spaces in GR and as far as I know, a metric induces a topology anyway?!

Fair enough. I would say I have a reasonable understanding of metric spaces, in terms of learning about open balls, open/closed sets, boundaries, homeomorphisms, etc. but I suspect this may not be quite deep enough?!
Yes, you'll need somewhat more: compactness, connectedness, completeness, continuity, uniform continuity, convergence of sequences, sequences of functions, etc.

What confuses me, is that whenever I read a text on GR, the author always starts by introducing spacetime as a 4D topological manifold. There is a mention of it requiring to have a topology defined on it with the Hausdorff property (i.e. two given distinct points in the set can be separated into disjoint neighbourhoods) and being second countable (i.e. the base of the topology is countable - I have to admit, I'm not particularly familiar with this), but I'm unsure as to the significance of this since one only considers metric spaces in GR and as far as I know, a metric induces a topology anyway?!
OK, let me try to explain. Consider a sphere in $\mathbb{R}^3$. We can measure the distance between two points on the sphere by drawing a straight line between the points and measuring the distance. This induces a metric on the sphere. But the metric is completely useless since it does not give us the actualy distance an inhabitant of the sphere might measure. The metric is still good for one thing: it gives us a "notion of closeness". So although the distance between two points on the sphere is not useful, we might still accurately say that two points are close or not so close. We can still talk about continuity and convergence using that metric. And it will work fine.

Of course we can consider a more physical metric on the sphere by consider the distance along "great arcs". But in GR this is not possible. There simply is NO physical meaningful metric. Spacetime is still a metric space, but the metric is not so useful. What is still useful is a degree of closeness on spacetime that tells us when two points are close, that tells us about convergence and continuity.

Topology does exactly that: topology takes away the metric and tries to talk merely about "degrees of closeness". Intuitively, we can see a topology as a set-values metric, where we say that a point $x$ is $G$-close to $y$ if $G$ is an open ball around $y$ that contains $x$. This is a sometimes useful thing to consider, but might be weird at first sight. But instead of talking about actual distances, we talk about the open sets which form "distances" somehow.

Anyway, Hausdorff is nothing more than ensuring that we cannot converge to more than one point.
Second countable is a very technical condition and is equivalent with the metrizability of the manifold. It ensures us a technical tool: the partition of unity which is very useful. Spaces that are not second countable tend to be very large, and can be restricted to second countable spaces anyway, so it's not a big loss.

FactChecker
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Defining a topology of open sets does not always give you a metric. The projection of the complex plane onto the real axis gives a simple, naturally occurring, example of a topological space that is not a metric space. Define d(z1, z2) = |Re(z1)-Re(z2)|. This is not a metric on the complex plane because d(z1, z2) = 0 does not imply z1=z2. But the open sets defined the obvious way does define a topology on C. Any non-trivial projection onto a metric space can give a similar example. You can think of many examples where you only want to consider only one aspect of distinct elements (like Re()) when you define "near". That gives you a topology of open sets that is not a metric.

There simply is NO physical meaningful metric.
Is this because GR is diffeomorphism invariant? What is non-physical about it?

opology does exactly that: topology takes away the metric and tries to talk merely about "degrees of closeness". Intuitively, we can see a topology as a set-values metric, where we say that a point xx is GG-close to yy if GG is an open ball around yy that contains xx. This is a sometimes useful thing to consider, but might be weird at first sight. But instead of talking about actual distances, we talk about the open sets which form "distances" somehow.
So is the usage of topology in GR mainly exploiting this property, that it provides information on the "degree of closeness" of given points on a manifold? Does defining a topology also specify how local coordinate patches should be "sewn together" to construct the manifold?

Is this because GR is diffeomorphism invariant? What is non-physical about it?
Ask yourself how you would define the distance between two points on a sphere. You would travel between the two points on the shortest route available and measure the distance travelled. When we attempt to do the same thing in GR, then this is impossible. The shortest route is not unique and gives different answers. Furthermore, since the speed of light is finite, it is unclear how to define the distance between two points for which one would have to travel faster than light.
For these reasons, a physical meaningful metric does not exist. There is still a metric that spews out numbers for any two points you input, but this number is rather meaningless.

So is the usage of topology in GR mainly exploiting this property, that it provides information on the "degree of closeness" of given points on a manifold? Does defining a topology also specify how local coordinate patches should be "sewn together" to construct the manifold?
Yes, topology can do that. This is also one of the advantages about topology, since it is not clear how to sew together spaces in metric spaces.

When we attempt to do the same thing in GR, then this is impossible. The shortest route is not unique and gives different answers.
Is this because the connection does not commute with itself?

Furthermore, since the speed of light is finite, it is unclear how to define the distance between two points for which one would have to travel faster than light.
Isn't the metric only defined locally though? It acts on two vectors in the same tangent space (at a single spacetime point).

Yes, topology can do that. This is also one of the advantages about topology, since it is not clear how to sew together spaces in metric spaces.
Is this one of the reasons why it is a useful notion in GR since then one knows (unambiguously) which coordinate patches are intersecting and how to patch these together?!

You'll need to be careful. In mathematics there are two very distinct notions of a "metric". In a metric space, the metric is a globally defined distance function. A (pseudo)-Riemannian metric is only locally defined and that is not what I'm talking about here.

Sewing together patches is definitely one way to define a new manifold, but it is rarely done. It is much easier to give a direct global definition.

You'll need to be careful. In mathematics there are two very distinct notions of a "metric". In a metric space, the metric is a globally defined distance function. A (pseudo)-Riemannian metric is only locally defined and that is not what I'm talking about here.
Ah ok. So when you were talking about the metric above you were referring to a globally defined metric then?! So topology is useful then to give a notion of nearness on a manifold since the (globally defined) metric is not a useful quantity. Does the topology of a (pseudo-) Riemannian manifold help to define a (pseudo-) Riemannian metric on the manifold?

Ah ok. So when you were talking about the metric above you were referring to a globally defined metric then?! So topology is useful then to give a notion of nearness on a manifold since the (globally defined) metric is not a useful quantity. Does the topology of a (pseudo-) Riemannian manifold help to define a (pseudo-) Riemannian metric on the manifold?
Yes, you cannot define the (pseudo-)Riemannian metric without the initial topology and smooth structure.

Yes, you cannot define the (pseudo-)Riemannian metric without the initial topology and smooth structure.
Is this because you won't know which points are in which neighbourhoods of other points and cannot define a tangent space to each point, hence can't define a quantity that measures distance between two "nearby" points?

Is this because you won't know which points are in which neighbourhoods of other points and cannot define a tangent space to each point, hence can't define a quantity that measures distance between two "nearby" points?
You want the Riemannian metric to be varying smoothly. That is: if two points are nearby, then then values of the Riemannian metric must not be too different. This is a continuity statement, and for that you need a global topology. And indeed, even the very notion of a tangent space requires a topological structure.

FactChecker
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Ask yourself how you would define the distance between two points on a sphere. You would travel between the two points on the shortest route available and measure the distance travelled. When we attempt to do the same thing in GR, then this is impossible. The shortest route is not unique and gives different answers.
If two paths give different distances, then one is less than the other and the other can not be a shortest distance. And for any two distinct points, there must be a strictly positive lower limit of the lengths of paths between the points. It seems that all the requirements of a metric can be met.
Furthermore, since the speed of light is finite, it is unclear how to define the distance between two points for which one would have to travel faster than light.
For these reasons, a physical meaningful metric does not exist.
I don't think that the concept of distance on a Riemann manifold depends on "traveling" from one point to the other at any particular speed unless the time dimension is part of the distance.

If two paths give different distances, then one is less than the other and the other can not be a shortest distance. And for any two distinct points, there must be a strictly positive lower limit of the lengths of paths between the points. It seems that all the requirements of a metric can be met.
It can't, the distance would be imaginary. This is because of the peculiarities of the spacetime which I didn't bother to go into. Even on flat spacetime, the metric would be

$$d(x,y) = (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 - y_3)^2 - (x_4 - y_4)^2$$

It is precisely the minus sign which screws things up. The result is physically meaningful: a distance of zero means that two points can be reached only by light, negative and positive distances are also physical phenomenon corresponding to massive particles and tachyons. Taking the square root would yield imaginary distances. That is the problem. A Cayley-Klein metric is possible but also doesn't satisfy the usual metric axioms.

I don't think that the concept of distance on a Riemann manifold depends on "traveling" from one point to the other at any particular speed unless the time dimension is part of the distance.
It's spacetime, so time is exactly part of the distance.

robphy
Homework Helper
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Global Structure of Spacetimes - Geroch & Horowitz (1979)

Space-time structure from a global viewpoint - Geroch (1971)
http://www.worldcat.org/title/general-relativity-and-cosmology-proceedings-of-the-international-school-of-physics-enrico-fermi-course-47-varenna-on-lake-como-30th-june-12th-july-1969/oclc/16214690

https://en.wikipedia.org/wiki/Spacetime_topology

The class of continuous timelike curves determines the topology of spacetime - Malament (1977)
http://dx.doi.org/10.1063/1.523436

A new topology for curved space–time which incorporates the causal, differential, and conformal structures - Hawking, King & McCarthy (1976)
http://dx.doi.org/10.1063/1.522874

So you can see topology as a way to do analysis in metric spaces without having the bothersome metric to screw things up.
That's wrong, not everyone topological space is metrizable

I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the additional structure of an open set of open subsets of these points), I'm really struggling to understand intuitively why this is needed, particularly in relation to manifolds used in physical applications such as GR?! I've heard explanations such as, a topology provides a primitive notion of geometry to a set without introducing additional structure such as a metric etc, and that it in some sense quantifies the "nearness" of points to one another (in the set) without having a notion of distance, in terms of neighbourhoods of points in the set, but these seem a little unsatisfactory to me.
Does one require a topology to be defined on a manifold such that the is a well-defined why to "stitch together" local coordinate patches to construct a global structure that is the manifold? Is it also required to define a notion of continuity on the manifold?
I think you need to start reading a good regular book on differential geometry. It is a bad idea to study geometry or any other math. subject by outlines and supplements which provide books in physics
for example all parts of this book https://www.amazon.com/dp/0387976639/?tag=pfamazon01-20
are simple to understand

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That's wrong, not everyone topological space is metrizable
For gods sake, can't I give some intuition without anybody complaining about rigor. Seriously, the OP wanted to know about the intuition for a topology, starting to whine about the intricacies of metrizability is not doing him any favours.

For the purpose of GR, all nonmetrizable spaces are pathological.

You want the Riemannian metric to be varying smoothly. That is: if two points are nearby, then then values of the Riemannian metric must not be too different. This is a continuity statement, and for that you need a global topology. And indeed, even the very notion of a tangent space requires a topological structure.
Ah ok.

Another thing that has confused me is, in what way does a topology quantify "nearness" of points in the set? Is it simply that the more subsets that two points both lie within, then the nearer they are to one another? I've read that points that are in some sense "near" remain "near" under deformations (e.g. the famous example of deforming a coffee cup into a doughnut), i.e. one can't "pass" subsets "through" one another?!

Ah ok.

Another thing that has confused me is, in what way does a topology quantify "nearness" of points in the set? Is it simply that the more subsets that two points both lie within, then the nearer they are to one another? I've read that points that are in some sense "near" remain "near" under deformations (e.g. the famous example of deforming a coffee cup into a doughnut), i.e. one can't "pass" subsets "through" one another?!
Here's my point of view of this. In a metric space, for every number $r$ we can say that $x$ is $r$-close to $y$ if $d(x,y)<r$. We can then formulate convergence of sequences as $x_n\rightarrow x$ iff for every $r$, the sequence is eventually $r$-close to $x$.

In a general topological space, we take an open neighborhood $V$ of $y$ and we say that $x$ is $V$-close to $y$ iff $x\in V$. We can then formulate convergence of sequences as $x_n\rightarrow x$ iff for every neighborhood $V$ of $x$, the sequence is eventually $V$-close to $x$.

This is how I'd quantify nearness in a topological space. It's far less intuitive than in a metric space, but it works.

I and that it in some sense quantifies the "nearness" of points to one another (in the set) without having a notion of distance, in terms of neighbourhoods of points in the set, but these seem a little unsatisfactory to me.
This MO post: http://mathoverflow.net/a/19156 should hopefully be exactly what you're looking for.