# Category:Axiom of Countable Choice

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This category contains results about Axiom of Countable Choice.

Definitions specific to this category can be found in Definitions/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 "Axiom of Countable Choice"

The following 44 pages are in this category, out of 44 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
- 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
- Countably Compact Metric Space is Compact
- Countably Compact Metric Space is Compact/Proof 1
- Countably Compact Metric Space is Compact/Proof 2

### D

### E

### I

### 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