Definition:Product Space

Topological Spaces

Let $\mathbb X = \left \langle {\left({X_i, \vartheta_i}\right)}\right \rangle_{i \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

Let $X$ be the cartesian product of $\mathbb X$:

$\displaystyle X := \prod_{i \in I} X_i$

Let $\mathcal T$ be the Tychonoff topology on $X$.

The topological space $\left({X, \mathcal T}\right)$ is called the direct product of $\mathbb X$.

Metric Spaces

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right)$ and $M_{2'} = \left({A_{2'}, d_{2'}}\right)$ be metric spaces.

Then we may define metrics on the cartesian product $A_{1'} \times A_{2'}$ in the same manner as the generalized Euclidean metric, as follows.

Let $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in A_{1'} \times A_{2'}$.

Let us define the following:

• $d_1 \left({x, y}\right) = d_{1'} \left({x_1, y_1}\right) + d_{2'} \left({x_2, y_2}\right)$
• $d_r \left({x, y}\right) = \left({\left({d_{1'} \left({x_1, y_1}\right)}\right)^r + \left({d_{2'} \left({x_2, y_2}\right)}\right)^r}\right)^{\frac 1 r}$
• $d_\infty \left({x, y}\right) = \max \left\{{d_{1'} \left({x_1, y_1}\right), d_{2'} \left({x_2, y_2}\right)}\right\}$.

Thus $\mathcal M = \left({A_{1'} \times A_{2'}, d_n}\right)$ with $d_n$ as variously defined above.

General Definition

The definition can be extended to the cartesian product of any finite number $n$ of metric spaces.

Let $M_{1'} = \left({A_{1'}, d_{1'}}\right), M_{2'} = \left({A_{2'}, d_{2'}}\right), \ldots, M_{n'} = \left({A_{n'}, d_{n'}}\right)$ be metric spaces.

Let $\displaystyle \mathcal M = \left({\prod_{i=1}^n \left({A_{i'}, d_{i'}}\right), d_n}\right)$, where the definition of $d_n$ is defined as:

• $\displaystyle d_1 \left({x, y}\right) = \sum_{i=1}^n d_{i'} \left({x_i, y_i}\right)$
• $\displaystyle d_r \left({x, y}\right) = \left({\sum_{i=1}^n \left({d_{i'} \left({x_i, y_i}\right)}\right)^r}\right)^{\frac 1 r}$
• $\displaystyle d_\infty \left({x, y}\right) = \max_{i=1}^n \left\{{d_{i'} \left({x_i, y_i}\right)}\right\}$

where $\displaystyle x = \left({x_1, x_2, \ldots, x_n}\right) \in \prod_{i=1}^n A_{i'}$ and $\displaystyle y = \left({y_1, y_2, \ldots, y_n}\right) \in \prod_{i=1}^n A_{i'}$.