Topological Equivalence of Product Spaces

If $ X $, $ Y $, and $ Z $ are topological spaces, the products $$ (X \times Y) \times Z $$ $$ X \times (Y \times Z) $$ $$ X \times Y \times Z $$ are all topologically equivalent. $$ (X \times Y) \times Z \cong X \times (Y \times Z) \cong X \times Y \times Z $$

This means that no matter how you group the spaces in a Cartesian product, the resulting topological space remains the same.

In other words, the Cartesian product of topological spaces is associative.

Note: This property simplifies the analysis of product spaces because it allows us to work with the product of multiple topological spaces without worrying about the order or grouping of the spaces involved.

A Practical Example

To illustrate the topological equivalence of products, consider some basic topological spaces like $\mathbb{R}$ (the real numbers with the standard topology) and $\mathbb{R}^2$ (the Cartesian plane with the product topology).

Let's take three copies of the space $\mathbb{R}$:

$X = \mathbb{R}$
$Y = \mathbb{R}$
$Z = \mathbb{R}$

Now, let's explore different ways to form products between these spaces:

Product (X×Y)×Z
First, we calculate $X \times Y$, which gives us the Cartesian plane $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$. Then, we take the product of $\mathbb{R}^2$ with $Z$, resulting in $\mathbb{R}^2 \times \mathbb{R}$. This product is a space of ordered triples $((x, y), z)$, where $x, y, z \in \mathbb{R}$, and can be identified with $\mathbb{R}^3$.
Product X×(Y×Z)
First, we compute $Y \times Z$, which again results in the Cartesian plane $\mathbb{R} \times \mathbb{R} = \mathbb{R}^2$. Then, we take the product of $X$ with $\mathbb{R}^2$, resulting in $\mathbb{R} \times \mathbb{R}^2$. This product is also a space of ordered triples $(x, (y, z))$, where $x, y, z \in \mathbb{R}$, and can be identified with $\mathbb{R}^3$.
Product X×Y×Z
Finally, we directly compute the Cartesian product of the three spaces $\mathbb{R}$, resulting in a space of ordered triples $(x, y, z)$, where $x, y, z \in \mathbb{R}$. This product, too, is equivalent to $\mathbb{R}^3$.

In all three cases, the resulting topological space is homeomorphic to $\mathbb{R}^3$.

The differences in parentheses or the order in which the products are calculated do not affect the final result, confirming that these products are topologically equivalent.

This example shows that no matter how you group the spaces, the final result is always the same topological space.