# Category:Axiom of Choice

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This category contains results about Axiom of Choice.

For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set.

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

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

- Axiom of Countable Choice (45 P)

## Pages in category "Axiom of Choice"

The following 80 pages are in this category, out of 80 total.

### A

### B

### C

- Cantor-Dedekind Hypothesis
- Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum
- Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum/Corollary
- Cardinals are Totally Ordered
- Cartesian Product of Subsets/Family of Nonempty Subsets
- 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
- Equivalence of Versions of Axiom of Choice/Formulation 1 implies Formulation 3
- Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1
- Equivalence of Versions of Axiom of Choice/Formulation 3 implies Formulation 1
- 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

- Hahn-Banach Theorem/Real Vector Space
- Hausdorff Maximal Principle
- Hausdorff Maximal Principle Implies Zorn's Lemma
- Hausdorff Maximal Principle is equivalent to Axiom of Choice
- Hausdorff Maximal Principle/Proof 1
- Hausdorff Maximal Principle/Proof 2
- Hausdorff Maximal Principle/Proof 3
- Hausdorff Paradox

### 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

### P

- Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Necessary Condition
- Product Space is Path-connected iff Factor Spaces are Path-connected
- Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Product Space is T3 1/2 implies Factor Spaces are T3 1/2
- Product Space is T3 iff Factor Spaces are T3
- Product Space is T3 iff Factor Spaces are T3/Product Space is T3 implies Factor Spaces are T3