Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Necessary Condition

Theorem

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$.

Let $\ds T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$.

Let $T$ be a completely Hausdorff space.

Then for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a completely Hausdorff space.

Proof

Suppose $T$ is a completely Hausdorff space.

As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.

Let $\alpha \in I$.

From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.

From completely Hausdorff property is hereditary, $T_\alpha$ is completely Hausdorff.

From Completely Hausdorff Space is Preserved under Homeomorphism, $\struct {S_\alpha, \tau_\alpha}$ is completely Hausdorff.

As $\alpha \in I$ was arbitrary, the result follows.

$\blacksquare$

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.