# Category:Axiom of Choice

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This category contains results about Axiom of Choice.

For every set of non-empty sets, we can provide a mechanism for choosing one element of each element of the set.

- $\displaystyle \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$

That is, one can always create a choice function for selecting one element from each element of the set.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### A

## Pages in category "Axiom of Choice"

The following 68 pages are in this category, out of 68 total.

### A

### B

### C

- Cantor-Dedekind Hypothesis
- Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum
- Cardinals are Totally Ordered
- Choice Function Exists for Well-Orderable Union of Sets
- Compact Subspace of Hausdorff Space is Closed/Proof 1
- Condition for Composite Mapping on Right
- Continuous Real Function Differentiable on Borel Set
- Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound

### D

### E

- Equivalence of Definitions of Nilradical of Ring
- Equivalence of Versions of Axiom of Choice
- Existence of Minimal Uncountable Well-Ordered Set/Proof Using Choice
- Existence of Non-Measurable Subset of Real Numbers
- Existence of Set with Singleton Intersections with Disjoint Collection
- Existence of Vector Space Bases implies Axiom of Choice

### F

### H

### I

- Inductive Construction of Sigma-Algebra Generated by Collection of Subsets
- Infinite Set has Countably Infinite Subset/Proof 1
- Infinite Set has Countably Infinite Subset/Proof 2
- Infinite Set has Countably Infinite Subset/Proof 3
- Irrational Numbers are Uncountably Infinite
- Irreducible Subspace is Contained in Irreducible Component