# Category:Projections

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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 $\ds \prod_{i \mathop = 1}^n S_i$ 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 $\ds S = \prod_{i \mathop = 1}^n S_i$** 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 the following 4 subcategories, out of 4 total.

## Pages in category "Projections"

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

### C

### F

### I

### P

- Paracompactness is Preserved under Projections
- Preimage of Element under Projection
- Product Topology is Coarsest Topology such that Projections are Continuous
- Projection from Cartesian Product under Chebyshev Distance is Continuous
- Projection from Metric Space Product with Euclidean Metric is Continuous
- Projection from Metric Space Product with P-Product Metric is Continuous
- Projection from Metric Space Product with Taxicab Metric is Continuous
- Projection from Product Category
- 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 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