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I've been trying to make sense recently of the usage of 'topological' in various fields of physics, and get sort of an intuition for what this means in context. This all boils down to my main question - if the usage of topology indicates working in a general topological space - does it make sense not to have a metric in physics? The specific and most important example I'm trying to get my head around is Topological Quantum Field Theory - I'm wondering how this formulation of QFT somehow works without metrics and how one can get an intuition for that being possible in a physical context.

Apologies if the question is somewhat vague, as this reflects my understanding at this point on this topic, but any very general insight would be much appreciated.

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The "topological" in "topological order" and the "topological" in "topological insulator" have different meanings.

The 'topological' in topological order means 'robust against ANY local perturbations'.

The "topological" in "topological insulator" means 'robust against some local perturbations that respect certain symmetry'. In fact the properties of symmetry-breaking order are also 'robust against some local perturbations that respect the symmetry'. In this sense, we may also call symmetry-breaking order "topological" (in the same sense we call topological insulator "topological").

There are two kinds of topology in math. The "topology" in "topological order" is directly related to the first kind of topology in mathematics, as in algebraic topology, homology, cohomology, tensor category, and topological quantum field theory. The "topology" in "topological insulator" is related to the second kind of topology in mathematics, as in mapping class, homotopy, K-theory, etc. The first kind of topology is algebraic, while the second kind of topology is related to the continuous manifold of finite dimensions. We may also say that the first kind of topology is "quantum", while the second kind of topology is "classical".

See Reconciling topological insulators and topological order

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