Definition:Product Space

Topological Spaces
Let $$(X_i)_{i \in I}$$ be a family of topological spaces where $$I$$ is an arbitrary index set.

Let $$X$$ be the cartesian product $$X := \prod_{i \in I} X_i$$.

For each $$i \in I$$ let $$\pi_i : X \to X_i$$ be the corresponding projection which maps $$(x_i)_{i \in I} \in X$$ to $$x_i$$.

The initial topology $$\mathcal{T}$$ on $$X$$ with respect to the family $$(\pi_i)_{i \in I}$$ is called the product topology on $$X$$.

The topological space $$(X, \mathcal{T})$$ is called the direct product of the $$(X_i)_{i \in I}$$.

$$\mathcal{T}$$ is generated by $$\mathcal{S} := \{ \pi_i^{-1}(U) | i \in I, U \subseteq X_i \text{ open}\}$$.

The basis $$\mathcal{S}^* := \left\{{\bigcap S : S \subseteq \mathcal{S} \text{ finite}}\right\}$$ (which is generated by $$\mathcal{S}$$) is called the natural basis of $$X$$.

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.

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


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

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

Relationship with Generalized Euclidean Metric
Let each of $$\left({A_{i'}, d_{i'}}\right)$$ be the real number line $$\R$$ under the usual metric.

Thus the Cartesian product $$\prod_{i=1}^n \left({A_{i'}, d_{i'}}\right)$$ is the $n$-dimensional real vector space $$\R^n$$.

Then the product space metrics as described here become the generalized Euclidean metrics.

Notation
The notation is awkward, because it is necessary to use a indexing subscript for the $$n$$ metric spaces contributing to the product, and also for the $$r$$th exponential that defines the metric itself.

Thus the "prime" notation on the $$n$$ metric spaces.