Filter on a Product Space Converges iff Projections Converge
Theorem
Let $(X_i)_{i \in I}$ be a family of non-empty topological spaces where $I$ is an arbitrary index set.
Denote with $\displaystyle X := \prod_{i \in I} X_i$ the corresponding product space.
Let $\mathcal F$ be a filter on $X$ and let $x \in X$.
Denote with $\operatorname{pr}_i : X \to X_i$ the projection from $X$ onto $X_i$.
Then $\mathcal F$ converges to $x$ iff for all $i \in I$ the image filter $\operatorname{pr}_i \left({\mathcal F}\right)$ converges to $x_i := \operatorname{pr}_i \left({x}\right)$.
Proof
- Assume first that $\mathcal F$ converges to $x$.
Let $i \in I$, then $\operatorname{pr}_i : X \to X_i$ is continuous.
Thus, $\operatorname{pr}_i \left({\mathcal F}\right)$ converges to $\operatorname{pr}_i \left({x}\right)$ as claimed.
- Assume now that for all $i \in I$, $\operatorname{pr}_i \left({\mathcal F}\right)$ converges to $\operatorname{pr}_i \left({x}\right)$.
Let $V \subseteq X$ a neighborhood of $x$.
We have to show that $V \in \mathcal F$.
Then there is a set $U$ from the natural basis of $X$ such that $x \in U \subseteq V$.
By the definition of the natural basis, there is a finite set $J \subseteq I$ such that $\displaystyle U = \bigcap_{j \in J}\operatorname{pr}_j^{-1} \left({U_j}\right)$ where $U_j \subseteq X_j$ is an open set for all $j \in J$.
Thus $U_j$ is an open neigborhood of $x_j$ for all $j \in J$.
Since $\operatorname{pr}_j \left({\mathcal F}\right)$ converges to $x_j$ this implies $U_j \in \operatorname{pr}_j \left({\mathcal F}\right)$ for all $j \in J$.
This implies $\operatorname{pr}_j^{-1} \left({U_j}\right) \in \mathcal F$ for all $j \in J$ by the definition of $\operatorname{pr}_j \left({\mathcal F}\right)$.
Since $J$ is finite, it follows that $\displaystyle U = \bigcap_{j \in J} \operatorname{pr}_j^{-1} \left({U_j}\right) \in \mathcal F$.
Remember that $U \subseteq V$, $\mathcal F$ being a filter thus implies that also $V \in \mathcal F$.
Thus $\mathcal F$ converges to $x$ as claimed.
$\blacksquare$
