# Relation between Product and Box Topology

## Product and Box Topology

### 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 $\pr_i$.

### Box Topology

Let $\family {\struct {X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces.

Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is:

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

Define:

- $\ds \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$

Then $\BB$ is a synthetic basis on $X$, as shown on Basis for Box Topology.

The **box topology** on $X$ is defined as the topology $\tau$ generated by the synthetic basis $\BB$.

## Relation between Product and Box Topology

### Product Topology

- Product Space is Product in Category of Topological Spaces where it is shown that the product topology is the topology on the Cartesian product of topological spaces that defines the categorical product in the category of topological spaces. The product topology is therefore important in a categorical sense.

- Product Topology is Coarsest Topology such that Projections are Continuous where it is shown that the Product topology is the coarsest topology on the cartesian product of topological spaces for which the projections are continuous.

#### Natural Basis of Product Topology

- Natural Basis of Product Topology where it is shown that the natural basis on $X$ is the set $\BB$ of cartesian products of the form $\ds\prod_{i \mathop \in I} U_i$ where:

- for all $i \in I : U_i \in \tau_i$
- for all but finitely many indices $i : U_i = X_i$

- Natural Basis of Product Topology of Finite Product where it is shown that the natural basis on $\ds X = \prod_{k \mathop = 1}^n X_k$ is:

- $\BB = \set{\ds\prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$

### Box Topology

- Basis for Box Topology where it is shown that the extension of the characterisation of the natural basis for the Product topology on a finite cartesian product to an arbitrary product is a basis and therefore the box topology is a topology.

- Box Topology may not form Categorical Product in the Category of Topological Spaces, the box topology on the cartesian product of topological spaces does not always give us the categorical product in the category of topological spaces.

- Box Topology may not be Coarsest Topology such that Projections are Continuous where it is shown that the box topology is not in general the coarsest topology on the cartesian product of topological spaces for which the projections are continuous.

- Box Topology contains Product Topology where it is shown that the box topology is a finer topology than the product topology.

- Box Topology on Finite Product Space is Product Topology where it is shown that the box topology and the product topology are the same topology on a finite cartesian product of topological spaces. So the box topology is only of interest for an arbitrary or infinite product of topological spaces.