Open sets in a topology produced by metrics.

Question

For each $i\in I$, $$d_i:X^2\to \Bbb R$$ is a metric and $\mathcal T$ is the coarsest topology on $X$ containing all topologies produced by the $d_i$ .

For $U\subseteq X$ we have: $$\forall x \in U,~\exists i\in I,~\exists r>0,~~B_x^{d_i}(r)\subseteq U$$ where $B_x^{d_i}(r)$ is the open ball with center $x$ and radius $r$.

Is $U$ open in $(X,\mathcal T)$?

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Edit:

Is $U$ open in some $(X,d_i)$?

Answer 1

Yes. For any $x\in X,i\in I$ and $r>0$, $B_x^{d_i}(r)\in\mathcal{T}_i$, the topology generated by $d_i$, and $\mathcal{T}_i\subseteq\mathcal{T}$, so $B_x^{d_i}(r)\in\mathcal{T}$ for each $x\in X,i\in I$ and $r>0$. In particular, each point of $U$ has a $\mathcal{T}$-open nbhd contained in $U$, so $U\in\mathcal{T}$.

Answer 2

For all $x \in U$, there exist $i \in I$ and $r_x>0$ such that $B_x^{d_i}(r_x) \subset U$, so $U= \bigcup\limits_{x \in U} B_x^{d_i}(r_x)$. But each $B_x^{d_i}(r_x)$ is open for the metric $d_i$, and a fortiori for $\mathcal{T}$ by definition.