# Category:Empty Set

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This category contains results about **the empty set**.

Definitions specific to this category can be found in Definitions/Empty Set.

The **empty set** is a set which has no elements.

That is, $x \in \O$ is false, whatever $x$ is.

## Subcategories

This category has the following 14 subcategories, out of 14 total.

### A

- Axiom of the Empty Set (1 P)

### E

- Empty Class (8 P)
- Empty Mapping (6 P)
- Empty Set is Closed (4 P)
- Empty Set is Unique (4 P)
- Empty Set is Well-Ordered (3 P)
- Empty Topological Space (2 P)
- Examples of Empty Sets (2 P)

### N

- Null Relation (4 P)

### U

- Union with Empty Set (3 P)

## Pages in category "Empty Set"

The following 87 pages are in this category, out of 87 total.

### C

- Cardinality of Empty Set
- Cartesian Product is Empty iff Factor is Empty
- Cartesian Product is Empty iff Factor is Empty/Family of Sets
- Cartesian Product of Family is Empty iff Factor is Empty
- Closure of Empty Set is Empty Set
- Collection of Sets Equivalent to Set Containing Empty Set is Proper Class
- Complement of Empty Set is Universe
- Complement of Universe is Empty Set
- Complement Union with Superset is Universe
- Complement Union with Superset is Universe/Corollary
- Countably Additive Function Dichotomy by Empty Set
- Cowen's Theorem/Lemma 2

### E

- Empty Intersection iff Subset of Complement
- Empty Intersection iff Subset of Relative Complement
- Empty Set as Subset
- Empty Set can be Derived from Comprehension Principle
- Empty Set Disjoint with Itself
- Empty Set forms Formal Language
- Empty Set from Principle of Non-Contradiction
- Empty Set is Closed
- Empty Set is Compact Space
- Empty Set is Countable
- Empty Set is Element of Power Set
- Empty Set is Element of Topology
- Empty Set is Initial Object
- Empty Set is Linearly Independent
- Empty Set is Nowhere Dense
- Empty Set is Null Set
- Empty Set is Open
- Empty Set is Open and Closed in Metric Space
- Empty Set is Open in Metric Space
- Empty Set is Open in Neighborhood Space
- Empty Set is Open in Normed Vector Space
- Empty Set is Ordinary
- Empty Set is Small
- Empty Set is Submagma of Magma
- Empty Set is Subset of All Sets
- Empty Set is Subset of Power Set
- Empty Set is Unique
- Empty Set is Well-Ordered
- Empty Set Satisfies Topology Axioms
- Equivalence Class is not Empty
- Equivalence of Formulations of Axiom of Empty Set

### I

- Identity of Power Set with Union
- If Set Exists then Empty Set Exists
- Image of Empty Set is Empty Set
- Image of Empty Set is Empty Set/Corollary 1
- Image of Empty Set is Empty Set/Corollary 2
- Infimum of Empty Set is Greatest Element
- Infimum of Power Set
- Intersection is Empty and Union is Universe if Sets are Complementary
- Intersection of Elements of Power Set
- Intersection with Complement
- Intersection with Complement is Empty iff Subset
- Intersection with Empty Set

### R

### S

- Set Consisting of Empty Set is not Empty
- Set Difference Equals First Set iff Empty Intersection
- Set Difference is Anticommutative
- Set Difference of Intersection with Set is Empty Set
- Set Difference with Empty Set is Self
- Set Difference with Intersection
- Set Difference with Self is Empty Set
- Set Difference with Superset is Empty Set
- Singleton Class of Empty Set is Supercomplete
- Subset of Empty Set
- Subset of Empty Set iff Empty
- Supremum of Empty Set is Smallest Element
- Swelled Class contains Empty Set
- Symmetric Difference of Equal Sets
- Symmetric Difference with Empty Set
- Symmetric Difference with Self is Empty Set