# Category:Surjections

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This category contains results about Surjections.

Definitions specific to this category can be found in Definitions/Surjections.

Let $S$ and $T$ be sets or classes.

Let $f: S \to T$ be a mapping from $S$ to $T$.

$f: S \to T$ is a **surjection** if and only if:

- $\forall y \in T: \exists x \in \Dom f: \map f x = y$

That is, if and only if $f$ is right-total.

## Subcategories

This category has the following 10 subcategories, out of 10 total.

### C

### E

### P

### S

## Pages in category "Surjections"

The following 70 pages are in this category, out of 70 total.

### C

- Cantor's Theorem
- Cardinality of Set of Induced Equivalence Classes of Surjection
- Cardinality of Set of Surjections
- Cardinality of Surjection
- Compactness is Preserved under Continuous Surjection
- Compactness Properties Preserved under Continuous Surjection
- Composite of Injection on Surjection is not necessarily Either
- Composite of Surjection on Injection is not necessarily Either
- Composite of Surjections is Surjection
- Composite of Three Mappings in Cycle forming Injections and Surjection
- Condition for Composite Mapping to be Identity
- Countability Axioms Preserved under Open Continuous Surjection
- Countable Compactness is Preserved under Continuous Surjection

### F

### I

- Identity Mapping is Surjection
- Image of Preimage of Subset under Surjection equals Subset
- Inclusion Mapping is Surjection iff Identity
- Injection from Finite Set to Itself is Surjection
- Injection has Surjective Left Inverse Mapping
- Inverse Image Mapping of Injection is Surjection
- Inverse of Surjection is Relation both Left-Total and Right-Total

### M

- Mapping from Finite Set to Itself is Injection iff Surjection
- Mapping is Injection and Surjection iff Inverse is Mapping
- Mapping is Surjection if its Direct Image Mapping is Surjection
- Mapping to Image is Surjection
- Mapping to Singleton is Surjection
- Mappings in Product of Sets are Surjections
- Mappings in Product of Sets are Surjections/Family of Sets

### O

### P

### S

- Second-Countability is Preserved under Open Continuous Surjection
- Sequential Compactness is Preserved under Continuous Surjection
- Set Finite iff Surjection from Initial Segment of Natural Numbers
- Sigma-Compactness is Preserved under Continuous Surjection
- Subset equals Image of Preimage iff Mapping is Surjection
- Subset equals Image of Preimage implies Surjection
- Successor Mapping on Natural Numbers is not Surjection
- Surjection from Countably Infinite Set iff Countable
- Surjection from Finite Set to Itself is Permutation
- Surjection from Natural Numbers iff Countable
- Surjection from Natural Numbers iff Countable/Corollary 1
- Surjection from Natural Numbers iff Countable/Corollary 2
- Surjection from Natural Numbers iff Right Inverse
- Surjection if Composite is Surjection
- Surjection iff Cardinal Inequality
- Surjection iff Epimorphism in Category of Sets
- Surjection iff Right Cancellable
- Surjection iff Right Inverse
- Surjection Induced by Powerset is Induced by Surjection
- Surjective Restriction of Inclusion is Identity
- Surjective Restriction of Real Exponential Function