# Topology in Physics

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How are topological spaces used in physics?

martinbn
How are topological spaces used in physics?
What do you know about topological spaces? And what do you know about physics?

What do you know about topological spaces? And what do you know about physics?
Quantitatively and conceptually, I know about as much as first year physics student physics-wise. Regarding topology, I hardly even know what it means.

fresh_42
Mentor
2021 Award
Quantitatively and conceptually, I know about as much as first year physics student physics-wise. Regarding topology, I hardly even know what it means.
Topology in its basics plays an important role in physics, simply because Hilbert spaces where physics mostly takes place are topological vector spaces.

WWGD
Gold Member
Topology is/provides, among other things, a framework to talk about continuity , and a way of compare properties of spaces. EDIT: You may want to consider whether standard Euclidean space, aka ##\mathbb R^4 ## is topologically the same as Space -Time 4-space. It is not, but maybe , e.g., @Orodruin can shed some more light on that.

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Spinnor
fresh_42
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2021 Award
Theres been some work in applying point set topology to black hole theory.

https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/blackhl.htm
Didn't convince me. A bit too simple and 3D for my taste, and partly wrong.
If it [EH] attaches to the exterior, then the known universe is topologically a closed set with respect to the black hole. If it [EH] attaches to the interior, then the known universe is topologically an open set.
It would only locally be closed or open. Additionally, the EH is hardly a real life hyperspace.

martinbn
Theres been some work in applying point set topology to black hole theory.

https://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/blackhl.htm
Is that it? I would hardly call it work applying point set topology to black holes. All it says is that the outside of the EH is an open set, if you include the boundary it is a closed set. Same for the interior. That is trivial. Even the definition of a manifold to describe any spacetime (black hole or not) needs more topology than that.

Spinnor
Gold Member
A Google image search sometimes gives interesting places to look,

https://room.eu.com/news/results-of-this-years-nobel-prize-in-physics-is-released
Edit, the Google search should have been for "topological spaces" but the search "topology in physics" might be of interest as well.

fresh_42
Mentor
2021 Award
Edit, the Google search should have been for "topological spaces" but the search "topology in physics" might be of interest as well.
Yes, but in this physical context "topology" means something else than the mathematical term.

Spinnor
Gold Member
Yes, but in this physical context "topology" means something else than the mathematical term.

My mistake. A Google search for the exact quote "topological spaces in physics" turns up one search item,

http://www.hep.caltech.edu/~fcp/math/HilbertSpace/HilbertSpace.pdf
A Google search "topological space on the arxiv preprint server" also turns up little but it did turn up this, "Theorists bridge space-time rips"

https://www.nature.com/polopoly_fs/...mns/topLeftColumn/pdf/491019a.pdf?origin=ppub

in which the term topological space appears.