# Category:Product Topology

This category contains results about **Product Topology**.

Definitions specific to this category can be found in Definitions/Product Topology.

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:

- $\ds \XX := \prod_{i \mathop \in I} X_i$

For each $i \in I$, let $\pr_i: \XX \to X_i$ denote the $i$th projection on $\XX$:

- $\forall \family {x_j}_{j \mathop \in I} \in \XX: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

The **product topology** on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

That is, $\tau$ is the topology generated by:

- $\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, U \in \tau_i}$

where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.

## Subcategories

This category has the following 4 subcategories, out of 4 total.

### N

### P

## Pages in category "Product Topology"

The following 41 pages are in this category, out of 41 total.

### B

### C

### F

### H

### I

### P

- Points in Product Spaces are Near Open Sets
- Product of Hausdorff Factor Spaces is Hausdorff/General Result
- Product Space Basis Induced from Factor Space Bases
- Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff
- Product Space is Homeomorphic to Product Space with Factors Commuted
- Product Space is Product in Category of Topological Spaces
- Product Space of Subspaces is Subspace of Product Space
- Product Topology is Coarsest Topology such that Projections are Continuous
- Products of Open Sets form Local Basis in Product Space
- Projection from Product Topology is Continuous
- Projection from Product Topology is Open
- Projection from Product Topology is Open and Continuous
- Projection from Topological Product is Identification Mapping

### U

- Uncountable Product of First-Countable Spaces is not always First-Countable
- Uncountable Product of Second-Countable Spaces is not always Second-Countable
- Uncountable Product of Separable Spaces is not always Separable
- Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact