Filter on Product of Hausdorff Spaces Converges iff Projections Converge
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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 if and only if for each $i \in I$ the image filter $\map {\pr_i} \FF$ converges.
Proof
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 $x$.
$\blacksquare$
Also see
- Filter on Product Space Converges iff Projections Converge, a more general result requiring the axiom of choice.