Dynamic Topology and the Impact of Expanding Spacetime on Subsets and Dimensions

[Mike2]

Just curious if anyone has ever studied what happens when a topology gains new members in the underlying set. How is it incorporated into the existing subsets whose union and intersection are included in the topology? It seems to me that assuming the universe expanded from a singularity, then more space with more time would add more elements to the underlying set, which is the universe as a whole. When we engrave a coordinate system on this topology (as with manifolds), Do the dimensions grow to incorporate the new elements of (spacetime?)? Are new subsets born which must be included? What? I'm not sure this question belongs here, but it seems it should be a consideration about the basic elements of an expanding spacetime. Shouldn't this be a consideration of quantum gravity? Any help is appreciated.

 

[selfAdjoint]

Just curious if anyone has ever studied what happens when a topology gains new members in the underlying set. How is it incorporated into the existing subsets whose union and intersection are included in the topology? It seems to me that assuming the universe expanded from a singularity, then more space with more time would add more elements to the underlying set, which is the universe as a whole. When we engrave a coordinate system on this topology (as with manifolds), Do the dimensions grow to incorporate the new elements of (spacetime?)? Are new subsets born which must be included? What? I'm not sure this question belongs here, but it seems it should be a consideration about the basic elements of an expanding spacetime. Shouldn't this be a consideration of quantum gravity? Any help is appreciated.

First of all the SIZE of the open sets that form a topology is not part of the topology definition, so they can grow or shrink without disturbing their union and intersection properties, which is what really defines the topology. A little teeny torus of Planck dimension and torus as big as the universe are both tori with the same topological proerties.

When you add a metric, then of course size, and change of size, enters in. But again, change of size by itself cannot change topology. Topology change is about "growing handles" and such things. That can happen at any size, and most of the topology-change research contemplates it happening at very small scales.

 

[Mike2]

First of all the SIZE of the open sets that form a topology is not part of the topology definition, so they can grow or shrink without disturbing their union and intersection properties, which is what really defines the topology. A little teeny torus of Planck dimension and torus as big as the universe are both tori with the same topological proerties.

I'm not sure I was referring to "size". But I am sure I was referring to "adding more elements to the underlying set". It seems necessary if the universe grew from a singularity, that more "points" must be added. If so, then does that automatically mean they are part of the topology? The topology was first defined in terms of existing elements and subsets of elements. Then another point arrives on the seen. Was it simply added to some pre-existing subset of the previously defined "topology"? Is it an additional subset in its own right? Is it not suppose to matter how one defines the subsets of a topology? I don't see where I've mentioned anything related to "size" in the above.

When you add a metric, then of course size, and change of size, enters in. But again, change of size by itself cannot change topology. Topology change is about "growing handles" and such things. That can happen at any size, and most of the topology-change research contemplates it happening at very small scales.

I'm not familiar enough with this subject to see how including unions and intersections of subsets gives rise to various kinds of genus. With a metric, a point is assigned coordinate. So what happens if one injects another point? Does the metric change? Is there an analytical continuation of "coordinates"? I don't know. But are these ideas the subject of a category of study

 

[Kea]

But I am sure I was referring to "adding more elements to the underlying set". It seems necessary if the universe grew from a singularity, that more "points" must be added. If so, then does that automatically mean they are part of the topology?

Mike2

Topology is more fundamental than the notion of a point. In axiomatic set theory, sets can be members (elements) of sets. A topology is a certain collection of sets. The study of topology starts from here. If you want to know what a point is, the clearest way is to think of it as a function from a one element set to some given set. This all happens in the category Set, of course. Then again, I suspect that you had in mind a more sophisticated notion of point, such as an event in a GR spacetime. This gets more complicated, but if you are interested in this question I think you'd like topos theory.

 

[Mike2]

Mike2
Topology is more fundamental than the notion of a point. In axiomatic set theory, sets can be members (elements) of sets. A topology is a certain collection of sets. The study of topology starts from here. If you want to know what a point is, the clearest way is to think of it as a function from a one element set to some given set. This all happens in the category Set, of course. Then again, I suspect that you had in mind a more sophisticated notion of point, such as an event in a GR spacetime. This gets more complicated, but if you are interested in this question I think you'd like topos theory.

I appreciate what you're trying to say. But I'm getting the impression that there has been no study of a dynamics of a topology, where points/elements/set-members might increase or decrease. I suppose this would be included in the dynamic of growing or shrinking manifolds. Has anyone ever heard of that? Perhaps that would be difficult since that would appear to be nothing more than a continuously changing coordinate system, right?

 

[Kea]

But I'm getting the impression that there has been no study of a dynamics of a topology, where points/elements/set-members might increase or decrease.

It sounds like you are referring to what Lawvere called variable sets. This is exactly what topos theory is about.

 

[Mike2]

It sounds like you are referring to what Lawvere called variable sets. This is exactly what topos theory is about.

Can you recommend an introductory text? Thanks.

 

[Kea]

Can you recommend an introductory text? Thanks.

A good recent text is:

Sheaves in Geometry and Logic: A first introduction to Topos Theory
S. Mac Lane, I. Moerdijk
Springer 1992

There are other good texts. Just have a look in a decent library. Warning: this stuff is never easy at first encounter, like anything worthwhile!

Cheers
Kea

 

[Chronos]

Differential geometry would be a good place to start. It is nearly impossible to understand set theory without it.

 

[Mike2]

A good recent text is:
Sheaves in Geometry and Logic: A first introduction to Topos Theory
S. Mac Lane, I. Moerdijk
Springer 1992
There are other good texts. Just have a look in a decent library. Warning: this stuff is never easy at first encounter, like anything worthwhile!
Cheers
Kea

As I understand it, sheaves are kind of complicated. What would be the prerequisites for this book?

I did some looking at Amazon.com, and I thought it might be easier for me to start with Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics) (Paperback) , by by J. Lambek, P. J. Scott, at:

https://www.amazon.com/gp/product/0521356539/?tag=pfamazon01-20

Could I get your opinion on whether this book would provide an introduction? Thanks.

 

[Hurkyl]

You might try looking at introductory category theory books. I'm interested in topoi too (but for their logical characteristics) (but I've been derailed from that for the moment), and I've been teaching myself out of these books I've checked out from the library (indefinitely, since nobody else seems to want to check them out. )


Categories for the Working Mathematician -- Saunders Mac Lane

Categories, Types, and Structures (An Introduction to Category Theory for the Working Computer Scientist) -- Asperti and Longo

Categories, Allegories. -- Freyd & Scedrov

in addition to Sheaves in Geometry and Logic.


I generally feel that studying from several textbooks is better than from just one. For example, when I understand something from Categores, Allegories, I really feel like I know it. But trying to understand that book without seeing the same topics presented in the other ones would be a nightmare!


Oh, I actually picked up a little bit about sheaves from Hartshorne's Algebraic Geometry, but that isn't exactly an easy text either.

 

[mccrone]

I'm getting the impression that there has been no study of a dynamics of a topology, where points/elements/set-members might increase or decrease. I suppose this would be included in the dynamic of growing or shrinking manifolds. Has anyone ever heard of that? Perhaps that would be difficult since that would appear to be nothing more than a continuously changing coordinate system, right?

Seems to me you have in mind questions of conservation laws and entropy - why is the creation of more void, an expanding universe, allowed? Why does the universe expand in short.

The "points" of the universe are Planck-scale. You could also ask why they don't expand as the Universe grows.

To start asking cosmologically realistic questions here, you need to remember that the Planck-scale defines both spacetime locatedness and also energy/mass density. That is the smallest scrap of "flat" spacetime, and also the maximum topological "buckling" of spacetime.

The general mathematical view of a point is very energy-less - and even timeless. So just defined in reference to a flat (ie: continuous) space.

A physically realistic notion of a point of spacetime has to be more complex.

From the above you will see that I'm talking about the expansion of the universe from a hot buckled point to a flat cold void. There is conservation of topology in that one thing is being exchanged for another. There is also dynamism in that one thing IS being exchanged for the other.

If you measure the universe in Planckian spacetime units, it is expanding. If you measure it in Planckian energy density units, it is flattening. The total number of Planckian units remains the same.

Cheers - John McCrone.

 

[Kea]

I did some looking at Amazon.com, and I thought it might be easier for me to start with Introduction to Higher-Order Categorical Logic (Cambridge Studies in Advanced Mathematics) (Paperback) , by by J. Lambek, P. J. Scott, at:
https://www.amazon.com/gp/product/0521356539/?tag=pfamazon01-20
Could I get your opinion on whether this book would provide an introduction? Thanks.

That book isn't that easy, either. Moreover, it is more for computer scientists than physicists. Another book you can get online is

Toposes, Triples and Theories
Michael Barr and Charles Wells
http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html

This is one of the easier books around. The advice to find a few different books to look at is good.