Finite Product Space is Connected iff Factors are Connected/General Case

Theorem

Let $I$ be an indexing set.

Let $\family {T_\alpha}_{\alpha \mathop \in I}$ be an indexed family of topological spaces.

Let $T = \ds \prod_{\alpha \mathop \in I} T_\alpha$ be the Cartesian space of $\family {T_\alpha}_{\alpha \mathop \in I}$.

Let $T = \ds \overline {\bigcup_{\alpha \mathop \in I} S_\alpha}$.

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Let $\tau$ be a topology on $T$ such that the subsets ${S'}_\alpha \subseteq \ds \prod T_\alpha$ where ${S'}_\alpha = \set {\family {y_\beta} \in T: y_\beta = x \beta \text { for all } \beta \ge \alpha}$ is homeomorphic to $S_{\alpha - 1} \times T_\alpha$.

Then $T$ is connected if and only if each of $T_\alpha: \alpha \in I$ are connected.

Proof

Let the Axiom of Choice be assumed.

Let $I$ be well-ordered.

Let $x = \family {x_\alpha} \in T$ be some arbitrary fixed element of $T$.

Let $S_\alpha = \set {\family {y_\beta} \in T: y_\beta = x \beta \text { for all } \beta \ge \alpha}$.

We have that $S_\alpha$ is homeomorphic to $S_{\alpha - 1} \times T_\alpha$.

Then from Finite Product Space is Connected iff Factors are Connected, $S_\alpha$ is connected if and only if $S_{\alpha - 1}$ is.

Let $\alpha$ be a limit ordinal.

Then:

$S_\alpha = \ds \paren {\bigcup_{\beta \mathop < \alpha} S_\beta}^-$

where $X^-$ denotes the closure of $X$.

So if each $S_\beta$ is connected for $\beta < \alpha$, it follows that $S_\alpha$ must likewise be connected, as $\family \gamma$ is nested.

Thus:

$T = \ds \overline {\bigcup_{\alpha \mathop \in I} S_\alpha}$ is connected.

$\blacksquare$

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.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Functions and Products