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2

The (Lie-)group $U(1)$ is the topological space $S^1$ (what we call a circle together with its standard open subsets) together with a rule how to multiply its points. In its representation as numbers in ${\mathbb C}$ with absolute value $1$, we have ${\mathrm e}^{{\mathrm i}\alpha}\bullet{\mathrm e}^{{\mathrm i}\beta}:={\mathrm e}^{{\mathrm ...


5

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 ...


4

It is simply false, at least written as it stands. The point is that the relation between the topology and the metric is more complicated than in the Riemannian case, where the geodesical balls form a basis of the topology$^1$. As a matter of fact, a (connected) Lorentzian smooth metric $g$ over the time-oriented smooth manifold $M$ does define a ...



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