Can you give me examples of topologies, that are not metric topologies, that are relevant in physics?
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$\begingroup$ Let us continue this discussion in chat. $\endgroup$– CuriousOneJan 22, 2016 at 10:44
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$\begingroup$ This question (v1) seems to be a list question. $\endgroup$– Qmechanic ♦Jan 22, 2016 at 11:32
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The strong dual of a Fréchet space (that is not a normed space) is not Fréchet and in particular not metrizable. The same can be said if we endow the dual with the weak-$*$ topology.
Therefore the space of distributions $\mathscr{S}'(\mathbb{R}^d)$ -- being the dual of the Fréchet but not Banach Schwartz space $\mathscr{S}(\mathbb{R}^d)$ -- is not a metrizable space when endowed with the usual topology(ies).
Distributions are pretty important in physics and appear in various contexts; however the topology of their space is in my knowledge seldom (or maybe never) used as a tool in physics.
Other non-metrizable topologies that are maybe more relevant in quantum physics applications are all the operator topologies excluding the norm topology (they are relevant to study convergence of operators, especially the strong and weak operator topology).
