Category:Examples of Use of Axiom of Countable Choice
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This category contains examples of use of Axiom:Axiom of Countable Choice.
Form 1
Let $\sequence {S_n}_{n \mathop \in \N}$ be a sequence of non-empty sets.
The axiom of countable choice states that there exists a sequence:
- $\sequence {x_n}_{n \mathop \in \N}$
such that $x_n \in S_n$ for all $n \in \N$.
Form 2
Let $S$ be a countable set of non-empty sets.
Then $S$ has a choice function.
Pages in category "Examples of Use of Axiom of Countable Choice"
The following 47 pages are in this category, out of 47 total.
C
- Complete and Totally Bounded Metric Space is Sequentially Compact
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 1
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 2
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 3
- Complete and Totally Bounded Metric Space is Sequentially Compact/Proof 4
- Completeness Criterion (Metric Spaces)
- Completeness Criterion (Metric Spaces)/Proof 1
- Completeness Criterion (Metric Spaces)/Proof 2
- Completion Theorem (Metric Space)
- Completion Theorem (Metric Space)/Lemma 3
- Construction of Outer Measure
- Countable Union of Countable Sets is Countable/Informal Proof
- Countable Union of Countable Sets is Countable/Proof 1
- Countable Union of Meager Sets is Meager
- Countably Compact Metric Space is Compact
- Countably Compact Metric Space is Compact/Proof 1
- Countably Compact Metric Space is Compact/Proof 2
D
I
L
M
S
- Second-Countable Space is Separable
- Separable Metacompact Space is Lindelöf/Proof 2
- Sequence of Implications of Metric Space Compactness Properties
- Sequentially Compact Metric Space is Compact
- Sequentially Compact Metric Space is Compact/Proof 3
- Sequentially Compact Metric Space is Second-Countable
- Sequentially Compact Metric Space is Separable
- Sequentially Compact Metric Space is Totally Bounded
- Sequentially Compact Metric Space is Totally Bounded/Proof 1
- Sequentially Compact Metric Space is Totally Bounded/Proof 2
- Sequentially Compact Space is Countably Compact
- Set of Finite Subsets of Countable Set is Countable/Proof 3