Filter on Product Space Converges iff Projections Converge

Theorem
Let $\left\langle{X_i}\right\rangle)_{i \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space.

Let $\operatorname{pr}_i: X \to X_i$ denote the projection from $X$ onto $X_i$.

Then $\mathcal F$ converges for all $i \in I$ the image filter $\operatorname{pr}_i \left({\mathcal F}\right)$ converges.

Sufficient Condition
Let $\mathcal F$ converge.

Then there is a point $x \in X$ such that $\mathcal F$ converges to $x$.

By Filter on Product Space Converges to Point iff Projections Converge to Projections of Point:
 * $\forall i \in I: \operatorname{pr}_i \left({\mathcal F}\right)$ converges to $x_i$

Thus
 * $\forall i \in I: \operatorname{pr}_i \left({\mathcal F}\right)$ converges.

Necessary Condition
Suppose that for each $i \in I$, $\operatorname{pr}_i \left({\mathcal F}\right)$ converges.

For each $i \in I$, let $S_i$ be the set of points to which $\operatorname{pr}_i \left({\mathcal F}\right)$ converges.

By our supposition, $S_i$ is non-empty for each $i \in I$.

By the axiom of choice, there is a point $x \in X$ such that:
 * $\forall i \in I: \operatorname{pr}_i\left({x}\right) \in S_i$

By Filter on Product Space Converges to Point iff Projections Converge to Projections of Point, $\mathcal F$ converges to $x$.

Also see

 * Filter on Product of Hausdorff Spaces Converges iff Projections Converge, a more restricted result that does not require the axiom of choice.