Schechter 2006: Kelley's cofinite proof yields BPI, not the Axiom of Choice

Kelley (1950) derived the Axiom of Choice from Tychonoff's theorem as follows. Given nonempty sets $X_a$ for $a \in I$, adjoin a point $\infty$ to each and topologise $X_a \cup \{\infty\}$ by the cofinite topology; each such space is compact, the sets $X_a$ are claimed to be closed, and the finite intersection property of the family $\{\pi_a^{-1}(X_a)\}$ then yields a point of the product, that is, a choice function.

Schechter (2006): that argument does not prove the Axiom of Choice. In the cofinite topology on $X_a \cup \{\infty\}$ the set $X_a$ is not closed when $X_a$ is infinite, since every nonempty open set is cofinite and therefore meets $X_a$. What Kelley's specialisation actually proves, and is equivalent to over ZF, is the Boolean prime ideal theorem.

The repair. Make the adjoined point isolated: topologise $X_a \cup \{\infty\}$ as the disjoint sum of $X_a$ with its cofinite topology and the one-point space $\{\infty\}$, so that the open sets are the unions of a cofinite subset of $X_a$ (or $\emptyset$) with a subset of $\{\infty\}$. This space is compact and $T_1$, and now $X_a$ genuinely is closed, its complement $\{\infty\}$ being open. With the repair the classical conclusion survives in the form: "every product of compact $T_1$ spaces is compact" is equivalent to the Axiom of Choice over ZF, while "every product of compact Hausdorff spaces is compact" is equivalent to BPI.

• Not proved in this library. Neither Tychonoff's theorem nor either equivalence is proved here.

• What would prove it. A ZF development of product topologies and compactness, plus the filter arguments of the ultrafilter lemma. Only the independence half, that BPI does not imply the Axiom of Choice, needs forcing; that half is Halpern and Lévy 1971: the Boolean prime ideal theorem does not imply the Axiom of Choice ‡. The rest is elementary once topology exists, which is why this item records a correction rather than a deep theorem.

• Why it matters here. The claim "Tychonoff implies AC" is standard, and the standard proof of it is the one Kelley gave. A library that repeats it verbatim would be asserting an equivalence with The Axiom of Choice ↗ on the strength of a false lemma, and would also be mis-pricing the Hausdorff case, which costs only The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter ↗. When the topology track reaches Tychonoff, this correction is what keeps the two statements and their two prices apart in The choice ledger: what costs the Axiom of Choice and what does not ↗.

• Conditional discipline. This item states an error and its repair, plus two ZF equivalences; only the accompanying strictness claim, that BPI is genuinely weaker, is an independence result and it is recorded conditionally in Halpern and Lévy 1971: the Boolean prime ideal theorem does not imply the Axiom of Choice ‡.