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the question is simple: is [itex] \mathcal{S}'\left(\mathbb{R}^3\right) [/itex] a first countable topological space ?

I have no idea, honestly. (The question has occured to me from a statement of Rafael de la Madrid in his PhD thesis when discussing the general rigged Hilbert space formalism. He says that even though the space of wavefunctions is assumed 1st countable, his antidual generally isn't. So I took the simplest case of a rigged Hilbert space: [itex] \mathcal{S}\left(\mathbb{R}^3\right)\subset \mathcal{L}^2 \left(\mathbb{R}^3\right)\subset \mathcal{S}^{\times}\left(\mathbb{R}^3\right) [/itex])

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# Is the space of tempered distributions 1st countable ?

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