Category:Examples of Use of Axiom of Choice
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This category contains examples of use of Axiom: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.
Pages in category "Examples of Use of Axiom of Choice"
The following 79 pages are in this category, out of 79 total.
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- 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
- Characterization of Paracompactness in T3 Space
- Characterization of Paracompactness in T3 Space/Lemma 13
- Characterization of Paracompactness in T3 Space/Lemma 8
- Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5
- Compact Subspace of Hausdorff Space is Closed/Proof 1
- Compactness Theorem/Proof using Ultraproducts
- Condition for Composite Mapping on Right
- Continuous Real Function Differentiable on Borel Set
- Countable Subset of Minimal Uncountable Well-Ordered Set Has Upper Bound
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- 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
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- 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