David Lipham

Real Analysis

7 December 2010

Many fundamental mathematical results fail in ZF

set theory without the Axiom of Choice.

“The Tychonoff Product Theorem concerning the

stability of compactness under formation of topological

product spaces may well be regarded as the single most

important theorem in general topology.” -H. Herrlich

Disaster

Intrinsic property of a set Finite sets

“small” (bounded), no limit points

In R, contain sup and inf as max and min

Process ends

Compact sets “next best thing to being finite”

“small”

An open cover of set E is an indexed family of open sets Gα 𝑖𝑛 ℜ 𝑠. 𝑡. 𝐸 ⊆ Gα𝛼 .

Set E is compact if and only if every open cover of E has a finite subcover. “If Gα is an open cover of E there exist finitely many 𝛼’s, say 𝛼 1, 𝛼 2, …

,𝛼 k such that 𝐸 ⊆ Gα i𝑘𝑖=1 = Gα 1 ,Gα 2 , … ,Gα k

.”

FIP (Finite intersection property) compactness: If each collection of closed subsets of X has FIP (the intersection of finitely many elements of the collection is non-empty), then the intersection of all elements of the collection is non-empty.

Heine-Börel

Closed subset of a compact set is compact

Continuous image of a compact set is compact

Theorem. Let 𝑋 and 𝑌 be compact spaces then 𝑋 × 𝑌 is compact. For finite products of compact sets, use mathematical induction. CC(fin) equivalent to countable products of finite or finite

discrete spaces (small subsets of compact spaces) compact. DC implies countable products compact spaces compact

note: countable products of compact spaces compact implies only CC.

Arbitrary (uncountably infinite) product?

Tychonoff’s Theorem (1935) states that the product of an arbitrary collection of compact topological spaces is compact. Proof requires establishing the product topology and

selecting a suitable version of compactness.

Definition. Given the (possibly infinite) Cartesian

product of the topological spaces Xi , indexed by 𝑖 ∈ 𝐼, and the canonical projections 𝜋𝑖 : X → Xi, the product topology on X is defined to be the coarsest topology (i.e. the topology with the fewest open sets) for which all the projections 𝜋𝑖 are continuous.

Proof required choosing an 𝑎𝛾from arbitrarily many

sets in order to define point (𝑎𝛼), thereby invoking the Axiom of Choice. Also, WOP

(𝐴𝐶 ⟹ 𝑇𝑦𝑐ℎ𝑜𝑛𝑜𝑓𝑓)

AC is sufficient, but is it necessary?

Some equivalent statements of the Axiom of Choice: Let C be a collection of nonempty sets. Then we can choose a

member from each set in that collection. There exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

For every family of non-empty pairwise disjoint sets there exists a set 𝑌 with 𝑌 ∩ 𝑋𝑖 = 1 for each 𝑖 ∈ 𝐼. “Given any nonempty set X whose members are pairwise disjoint sets,

there exists a set Y consisting of exactly one element taken from each set belonging to X.”

For each family (𝑋𝑖)𝑖∈𝐼 of non-empty sets 𝑋𝑖, the product set 𝑋𝑖𝑖∈𝐼 is non-empty.

Let 𝑋𝑖 𝑖∈𝐼 be an indexed family of nonempty sets.

Choose element 𝑝 such that 𝑝 ∉ 𝑋𝑖𝑖∈𝐼 .

Define 𝑌𝑖 = 𝑋𝑖 ∪ 𝑝 with cofinite topology 𝜏𝑖 = ∅, 𝑌𝑖 , 𝑋𝑖 , 𝑝

The cofinite topology is a set of all cofinite subsets (subsets of 𝑌𝑖 whose complement in 𝑌𝑖 is finite) as open sets, along with the empty set.

𝑌𝑖 is compact in 𝜏𝑖 since the number of open sets is finite, and by Tychonoff’s Theorem 𝑌 = (𝑌𝑖 , 𝜏𝑖𝑖∈𝐼 ) is compact.

The projection 𝜋𝑗: 𝑋𝑖𝑖∈𝐼 ⟶ 𝑋𝑗

𝑋𝑖 (the compliment of open singleton *𝑝+) , is closed in 𝑌𝑖

𝑋𝑖 and 𝑝 disjoint assures the compliment of *𝑝+ is 𝑋𝑖

And 𝜋𝑗 is continuous. Therefore

𝜋𝑗−1 𝑋𝑗 = 𝑋𝑗 × (𝑌𝑖 , 𝜏𝑖

𝑖≠𝑗

)

is a (nonempty) closed subset of 𝑌.

Note: 𝜋𝑗−1 𝑋𝑗 represents all possible preimages with 𝑋𝑗 in the 𝑗𝑡ℎ

position (disregard the other indices). The order of coordinates may not be preserved but all preimages are topologically homeomorphic.

𝑇𝑦𝑐ℎ𝑜𝑛𝑜𝑓𝑓 ⟹ 𝐴𝐶

For any finite subset Λ of 𝐼, we can define a point in the intersection

𝜋𝑗−1 𝑋𝑗

𝑗∈Λ

by 𝑋𝑗𝑗∈Λ

× (𝑌𝑖 , 𝜏𝑖𝑖∈𝐼\Λ

) = 𝑥𝜆 × 𝑝∗

(𝑝 at every coordinate position 𝑖 ∈ 𝐼\Λ) Note: Selecting 𝑝∗ did not require AC (!) since 𝑝 is a member of every 𝑌𝑖 by construction.

This means that a family 𝜋𝑗−1 𝑋𝑖 | 𝑗 ∈ Λ (closed subsets of

compact space 𝑌) has FIP.

𝑇𝑦𝑐ℎ𝑜𝑛𝑜𝑓𝑓 ⟹ 𝐴𝐶

By compactness of 𝑌, the entire intersection

𝜋𝑖−1 𝑋𝑖

𝑖∈𝐼

ℎ𝑎𝑠 𝐹𝐼𝑃 ≠ ∅

This intersection denotes 𝑋𝑖𝑖∈𝐼 , and we have shown AC.

∎

𝑇𝑦𝑐ℎ𝑜𝑛𝑜𝑓𝑓 ⟹ 𝐴𝐶

Variants of compactness equivalent in ZFC separate in ZF

Pseudometric Space Compactness in ZF: (CC sufficient)

“It is a hopeless endeavor, doomed to failure to attempt to prove the Tychonoff product theorem without invoking some form of the axiom of choice…. It is my feeling, however, that the definition of compactness relative to which the Tychonoff theorem is unprovable without the axiom of choice is, from the point of view of topological analysis and the theory of rings of continuous functions, unnatural and unsuitable.” - W.W. Comfort

Topological Space Compactness in ZF

Ex. Hausdorff spaces, complete regularity

(spaces which can be embedded in a product

of copies of the closed unit interval), etc

Ultrafilter-compact (Bourbaki)

Comfort-compact

Alexandroff-Urysohn-compact

Heine-Borel-compact

BPIT

Depending on the topological space and

chosen form of compactness, product

compactness holds in ZF – provided that AC

is replaced by some (relatively consistent)

alternative axiom.

Weak forms of AC (PIT, DC, CC,UFT, Dedekind-

finite sets are finite) offer bright spots, as

previously hinted.