Filter on Product of Hausdorff Spaces Converges iff Projections Converge

Theorem
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces where $I$ is an arbitrary index set.

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

Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$.

Let $\FF \subset \powerset X$ be a filter on $X$.

Then $\FF$ converges for each $i \in I$ the image filter $\map {\pr_i} \FF$ converges.

Sufficient Condition
Let $\FF$ converge.

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

By Filter on Product Space Converges to Point iff Projections Converge to Projections of Point:
 * $\forall i \in I: \map {\pr_i} \FF$ converges to $x_i$.

Thus, for each $i \in I$, $\map {\pr_i} \FF$ converges.

Necessary Condition
For each $i \in I$, let $\map {\pr_i} \FF$ converge.

Since $X_i$ is a Hausdorff space for each $i \in I$, this implies that $\map {\pr_i} \FF$ converges to exactly one point for each $i \in I$.

For each $i$, let $x_i$ be the point in $X_i$ to which $\map {\pr_i} \FF$ converges.

Then by the definition of the product space, $x$ is an element of $X$.

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

Also see

 * Filter on Product Space Converges iff Projections Converge, a more general result requiring the axiom of choice.