Filter on Product Space Converges iff Projections Converge

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Theorem

Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological 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 all $i \in I$ 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

$\forall i \in I: \map {\pr_i} \FF$ converges.


Necessary Condition

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

For each $i \in I$, let $S_i$ be the set of points to which $\map {\pr_i} \FF$ 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:

$\forall i \in I: \map {\pr_i} x \in S_i$

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

$\blacksquare$

Axiom of Choice

This theorem depends on the Axiom of Choice.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.


Also see