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**huberscher** Prove this statement or the opposite:

Consider the set $\displaystyle \mathbb{R}^{\mathbb{N}} := \{ (x_n)_{n \in \mathbb{N}} : x_n \in \mathbb{R} \forall n \in \mathbb{N} \} $ . Define $\displaystyle U_{y_1,...,y_m}(r):= \{ (x_n)_n \in \mathbb{R}^{\mathbb{N}} : |x_i - y_i| < r \text{ for every } i=1,...,m \} $ .

The collection $\displaystyle B:= \{ U_{y_1,...,y_m}(r) : y_1 ,...,y_m \in \mathbb{R}, m \in \mathbb{N},r>0 \}$ is a base for a topology on $\displaystyle \mathbb{R}^\mathbb{N} $ .