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Consider the unit simplex $S_d:=\{x \in [0, \infty)^d: \Vert x\Vert_1=1\}$ and denote by $\mathring{S}_d$ its interior. Then define the set $S_d'=\mathring{S}_d\cup\{{e_j}, j=1, \ldots, d\}$, where $e_j$ denotes the $j$-th vertex, i.e. the vector having unit $j$-th component and all the other components equal to zero. Is $S_d'$ a $G_\delta$-subset of $S_d$?

My guess is: yes, since it is the union of an open set (the interior) and the (closed) singletons $\{e_j\}$, all of which should be $G_\delta$, whence we conclude that the union is $G_\delta$ too, right?

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