## Topology terminology question

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lgw
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### Topology terminology question

I'm confused about terminology relating to Topology, and can't find anything in Wikipedia to set me straight (as is often the case, the articles assume you know too much about the field), so maybe you guys can help.

Between terms like compact, complete, and bounded, I'm looking for the right term for the concept "there exists some maximum distance that no two points can exceed" in the general sense across manifolds and metrics (from a finite Euclidian region, to a torus, to a crazy metric that just works that way). I'm also looking for the right term for the informal idea that "if you go far enough in a straight line, you might get back to where you started", one kind of manifolds that meet the previous definition. Reading the various WP articles, I get lost in all the other terminology I don't know.

Any generous soul want to take mercy on my confusion, and maybe explain any of compact, complete, and bounded that aren't the ideas above? I'm sure the distinction between "0 < x < 1", and "0 <= x <= 1" is tied up in all of this too, but I can't quite make sense of how it all gets expressed.
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Tirian
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### Re: Topology terminology question

lgw wrote:Between terms like compact, complete, and bounded, I'm looking for the right term for the concept "there exists some maximum distance that no two points can exceed" in the general sense across manifolds and metrics (from a finite Euclidian region, to a torus, to a crazy metric that just works that way).

The region would be a bounded region, the metric would be a bounded metric.

I'm also looking for the right term for the informal idea that "if you go far enough in a straight line, you might get back to where you started", one kind of manifolds that meet the previous definition. Reading the various WP articles, I get lost in all the other terminology I don't know.

There are no (intrinsic) "straight lines" in general topology.

lgw
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### Re: Topology terminology question

Tirian wrote:
lgw wrote:Between terms like compact, complete, and bounded, I'm looking for the right term for the concept "there exists some maximum distance that no two points can exceed" in the general sense across manifolds and metrics (from a finite Euclidian region, to a torus, to a crazy metric that just works that way).
The region would be a bounded region, the metric would be a bounded metric.

And the torus? Is there a single term that encompasses the concept?

Just to confirm the metric, let's say we had a plane where we measure distance as:

distance = sqrt(x2 + y2), where that's <= 1; otherwise
distance = 2 - sqrt(1/(x2 + y2))

That's a bounded metric? (Sorry if that's overly informal, I hope one can understand what I'm getting at.)

Tirian wrote:
lgw wrote:I'm also looking for the right term for the informal idea that "if you go far enough in a straight line, you might get back to where you started", one kind of manifolds that meet the previous definition. Reading the various WP articles, I get lost in all the other terminology I don't know.
There are no (intrinsic) "straight lines" in general topology.

Geodesic? Some other term? Help? The general concept that include a sphere, a torus, and any other sort of surface where it's possible that if I start driving my car along, turning the wheel neither left nor right, I could end up back where I started, though possibly upside down (or however one extends the analogy to N dimensions, and possibly turns inside out in the process).
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Tirian
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### Re: Topology terminology question

A) Yes. If the range of the metric is bounded, then the metric is bounded.

B) *I* understand what you mean. But general topology only knows about "closeness" and metric topologies only add in a concrete notion of distance. There is no "direction". It's more of a linear algebra concept to ask if the mapping (c \mapsto cv: R \rightarrow V) is not injective for some or all vectors v (and if there is a name for this property, I've never heard it).

gmalivuk
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### Re: Topology terminology question

So don't worry about direction and use distance. It seems lgw is asking if there's a (more concise) term for a space with closed geodesic paths.
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Tirian
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### Re: Topology terminology question

Maybe there is a term in the field, but it is not in my 2-semester textbook. Given that the standard explanation for why a sphere isn't homeomorphic to the torus is "because every loop on a sphere can be contracted to a point but there are loops on the torus that cannot be contracted to a point", topologists don't always seem to be keen on notational shorthand.

lgw
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### Re: Topology terminology question

Sorry, let me ask the questions a different way - I'm probably asking in a silly way.

1) "What do a sphere, a torus, and a Klein bottle all have in common (but a plane doesn't)?"

2) "What do those things, a bounded region, and a manifold with a bounded metric all have in common?"

The latter is "bounded", right? all the points lying within some fixed distance of another? And does "closed" include all the (1)ish things? Can there be a closed but unbounded space?

Thanks again for explaining.
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somehow
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### Re: Topology terminology question

This does not address all of the questions you raise in your last post, but it addresses some of them and might help clarify your thinking about the terms in question.

The term "closed" is relative: it applies to a subset of a larger space. It doesn't make sense to ask whether a space is closed in and of itself. For example, the half-open interval [0,1) is not closed when viewed as a subset of the real line, but it is closed when viewed as a subset of, say, the interval (-1,1).

Closed and bounded are unrelated, in the sense that we can find examples of subsets of topological spaces for every combination of {closed, not closed} and {bounded, not bounded}. In fact, we can find subsets of all four types just within the real line.
[0,1] as a subset of the real line is closed and bounded, and its complement (-infinity,0) U (1,infinity) is neither closed nor bounded.
(0,1) as a subset of the real line is bounded but not closed, and its complement (-infinity,0] U [1,infinity) is closed but not bounded.
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notzeb
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### Re: Topology terminology question

lgw wrote:Sorry, let me ask the questions a different way - I'm probably asking in a silly way.

1) "What do a sphere, a torus, and a Klein bottle all have in common (but a plane doesn't)?"

2) "What do those things, a bounded region, and a manifold with a bounded metric all have in common?"

The latter is "bounded", right? all the points lying within some fixed distance of another? And does "closed" include all the (1)ish things? Can there be a closed but unbounded space?

Thanks again for explaining.
For 1), a good answer is "compact" (some manifold people call this "closed", which is incredibly confusing, since "closed" has a totally different meaning in topology). For 2), I don't think there is such a good answer, because for instance the open interval (0,1) is bounded, but is diffeomorphic to the real line, which doesn't have the property you are looking for, but if you include the metric as part of the data of your space, then the word you are looking for is "bounded".

A property which you might like is the following: every continuous function from this space to the reals has bounded image. A space with this property is necessarily bounded in every continuous metric, since for every point b in your space, the function taking point p to the distance from b to p is continuous in p, and thus bounded. For manifolds, this property turns out to be the same as being compact (and in general, every compact space has this property, but some topological spaces having this property might not be compact).
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Forest Goose
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### Re: Topology terminology question

This isn't deeply helpful, but there is the notion of induced lines in metric spaces -

Let (X, d) be a metric space and x, y, z in X; say y is between x and z, written <xyz> , if d(x, z) = d(x, y) + d(y , x). The line induced by x and z is all points y so that <abc> where {a, b, c} = {x, y, z}.

There is no concept of "direction", of course - there are a few ways that you could try and force some notion of this wrt curves travelling along such lines, but I'm not sure if it would useful your purposes, or not.

Most of what I've read about these concepts seems to be regarding finite geometry or probabilistic metric spaces, and not really in the directions of things you're discussing - and it's not an area I know a lot about, so I have no good suggestions beyond just the definition.
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doogly
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### Re: Topology terminology question

Have you looked at the book Counterexamples in Topology? It is really helpful with the definition zoo, giving concrete examples of spaces with some and without other properties.

Like, can you show me a space that is bounded, but not closed? Closed, but not bounded? And bunches with metrization.
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lgw
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### Re: Topology terminology question

Thanks everyone - this will take me some time to think through. I started with this question just looking for terms for possible shapes of the universe (the kind where you might see your location in a good enough telescope vs not), but one wiki walk later, I know far less than I started with!

notzeb wrote:For 1), a good answer is "compact" (some manifold people call this "closed", which is incredibly confusing, since "closed" has a totally different meaning in topology).

And "compact" just means "closed and bounded", right? "The unit circle and the points inside it" is compact, right? I keep wanting to say "cyclic" here, but that implies the time dimension (at least to me).

Forest Goose wrote:This isn't deeply helpful, but there is the notion of induced lines in metric spaces -

Let (X, d) be a metric space and x, y, z in X; say y is between x and z, written <xyz> , if d(x, z) = d(x, y) + d(y , x). The line induced by x and z is all points y so that <abc> where {a, b, c} = {x, y, z}.

Wait, did you mean if d(x, z) = d(x, y) + d(y , z) ? Cause otherwise I'm totally lost here. And this clause is defining "between", right?

If I followed that, then the induced line isn't unique for some metrics? E.g, a taxicab metric. (Maybe that's what you mean about direction - not following that so much) Though it still is useful for the concept I was getting at, even then.
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Forest Goose
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### Re: Topology terminology question

Yes, that should be a z not an x

I'm not sure what you mean about uniquness, the line through two points isn't unique with the usual euclidean metric either - let L be the line through x and y, let u and v be on L and not x or y, then the line through u and v is also L. Or were you meaning something different?

The problem with direction is that there is no general sense of being able to say "z comes after y" to indicate that you are moving in the direction of y to z (as in pointing that way). You could start at a point x, then consider something like y -> z as defined by <xyz>, but I'm not sure that that would have all the properties you would want - nor does it encompass the entire line, even if so.
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gmalivuk
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### Re: Topology terminology question

lgw wrote:And "compact" just means "closed and bounded", right?
No, they just happen to be equivalent in Rn.
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Forest Goose
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### Re: Topology terminology question

Though, a subset of a uniform space is compact iff complete and totally bounded - this stronger version of the Heine Borel Theorem, thus, applies to all metric spaces and topological vector spaces.

For metric spaces this means:

- All cauchy sequences converge, and
- For every real d, there is a finite open cover by sets of diameter less than d.
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### Re: Topology terminology question

lgw wrote:And "compact" just means "closed and bounded", right?

No, the Heine-Borel Theorem says they are equivalent in Rn. A set S is compact if for every open cover of S can be reduced to a finite subcover. The key here is that no matter how big the original cover is I can always extract a finite number of open sets from that cover, and it still covers the set. Notice this must apply for any possible cover. For example for the open interval S = (0,1) then the cover C = {(0,n/(n+1) for n>=1} does cover the interval, but I will always need infinitely many of the sets in C to cover it, so it's not compact. Also notice I can easily cover (0,1) with a finite open cover, say C = {(0,1)} to be really show how trivial it is to construct. The point is you have to take into account every possible open cover, not just a chosen one.
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Tirian
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### Re: Topology terminology question

lgw wrote:If I followed that, then the induced line isn't unique for some metrics? E.g, a taxicab metric. (Maybe that's what you mean about direction - not following that so much) Though it still is useful for the concept I was getting at, even then.

If by "induced line [segment]", you mean {z \in X | d(x,z) + d(z,y) = d(x,y)}, then yes, it is not what you would perceive to be a model of "straightness" in every metric. As you note, it is a rectangle in the taxicab metric, and in the discrete metric it's just the doubleton {x, y}.

Forest Goose
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### Re: Topology terminology question

No, they're not "straight", but I wouldn't expect them to be.

The real defect, in terms of your question, is that they won't loop around (like a path on a sphere) since <xyx> requires d(x, x) = 2d(y, x), which can't happen unless x = y. However, it may be of interest to ask for when there is a sequence xn so that x0 = x, <xnxn+jxn+k> for all n and k >= j >= 0, and xn converges to x - I think that should replicate the notion to an extent - at least at first pass.*

That said, there are better ways to go about lines - like attaching a group of "translations" to the space and looking at how that works (you could impose whatever other "linelike" constraints you might want on that, algebraically); depending on what exactly you need, there are, most likely, numerous other ways to do it. The bottom line is that these are all just concepts attached to concepts, there is no "right" notion anymore so than that it is right to say an open ball, in some weird metric, is the right choice of "round" because it is a ball, what matters is what they let you say and what it is you are trying to say with it (not to say it is not good to draw inspiration, but the important point is not, "is it straight?", but, "can I do with it what I need to do with straightness?").

*Or just d(x0, xn) = d(x0, x1) +... d(xn-1, xn) and xn converges to x; you'd end up asking something like this with the notion of translations, just with indices - or doing something that gets iterates and limits ever smaller actions till you get a "line", then looking at that - or some such.
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gmalivuk
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### Re: Topology terminology question

Does 'geodesic' not capture the meaning we want? It's my understanding that a geodesic is locally a distance-minimizer, but that places no restrictions on the endpoints not being able to loop back around.
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### Re: Topology terminology question

It does, but it never hurts to have additional notions at hand, nothing is one size fits all and more never hurt (Rings are good till you need a lattice, even if both have two operations on them).

*Too, not all metric space are manifolds and not all topological manifolds are metric, so geodesics aren't going to work everywhere - and some notions could, probably, be thought up that would work in spaces that are neither metric nor manifold.
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gmalivuk
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### Re: Topology terminology question

Yeah, but if you're talking about distances, you're already in a metric space. (I meant specifically your induced line account, rather than other "line-like" ideas that have been discussed.)
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notzeb
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### Re: Topology terminology question

Back to the original question: do geodesics tend to eventually come back to their starting points?

If a Riemannian manifold is compact, then so is its unit tangent bundle (the unit tangent space at any point is a sphere). By Liouville's theorem, the geodesic flow preserves some natural measure (called the kinematic measure or the Liouville measure: see this wiki page) on the unit tangent bundle, and by compactness the unit tangent bundle will have a finite total volume with respect to this measure. Then by Poincaré's recurrence theorem (see another wiki page) almost all geodesics eventually come back arbitrarily close to where they started (and pointing in almost the same direction) infinitely many times.

Unfortunately, we don't actually live in a Riemannian manifold: we live in a Lorentzian manifold (maybe). Now the tangent space (and the possible geodesics) divide into three classes: lightlike, spacelike, and timelike. We could make a unit-spacelike-tangent bundle or a unit-timelike-tangent bundle, but neither of these would be compact (at any given point, we get hyperboloids instead of spheres). There is no concept like a unit-lightlike-tangent bundle at all, since all lightlike vectors have "length" 0, but we could take the projectivization of the lightlike tangent bundle (this corresponds to ignoring redshifts and blueshifts), and that would be compact if the original Lorentzian manifold was compact (at every point, we would be taking the projectivization of a lightcone, which is a sphere). I don't know if there is any analogue of the Liouville theorem in that case... maybe someone who actually studies physics or geometry could answer this.
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### Re: Topology terminology question

gmalivuk wrote:Yeah, but if you're talking about distances, you're already in a metric space. (I meant specifically your induced line account, rather than other "line-like" ideas that have been discussed.)

For induced lines, yes; but that relation is not far from something more general, and it's the same approach as attaching a group - or other like things. The idea of limiting small steps is pretty general and can be easily repurposed for more general purposes.
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