# Category:Projections

Jump to navigation
Jump to search

This category contains results about Projections in the context of Mapping Theory.

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

Let $S_1, S_2, \ldots, S_j, \ldots, S_n$ be sets.

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the Cartesian product of $S_1, S_2, \ldots, S_n$.

For each $j \in \set {1, 2, \ldots, n}$, the **$j$th projection on $\displaystyle S = \prod_{i \mathop = 1}^n S_i$** is the mapping $\pr_j: S \to S_j$ defined by:

- $\map {\pr_j} {s_1, s_2, \ldots, s_j, \ldots, s_n} = s_j$

for all $\tuple {s_1, s_2, \ldots, s_n} \in S$.

## Subcategories

This category has only the following subcategory.

## Pages in category "Projections"

The following 34 pages are in this category, out of 34 total.

### C

### E

### F

### P

- Paracompactness is Preserved under Projections
- Preimage of Element under Projection
- Projection from Cartesian Product under Chebyshev Distance is Continuous
- Projection from Product Category
- Projection from Product of Family is Injection iff Other Factors are Singletons
- Projection from Product of Family is Surjective
- Projection from Product Topology is Continuous
- Projection from Product Topology is Open
- Projection from Product Topology is Open and Continuous
- Projection is Epimorphism
- Projection is Epimorphism/General Result
- Projection is Injection iff Factor is Singleton
- Projection is Injection iff Factor is Singleton/Family of Sets
- Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition
- Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition
- Projection is Surjection
- Projection is Surjection/Family of Sets
- Projection is Surjection/General Version
- Projection of Complement Contains Complement of Projection
- Projection on Cartesian Product of Modules
- Projection onto Ideal of External Direct Sum of Rings