Arbitrary Topological Products of Topological Spaces

Arbitrary Topological Products of Topological Spaces

Recall that if $X$ and $Y$ are both topological spaces and $X \times Y$ denoted the Cartesian product (or simply just product) of these two spaces, then the product topology on $X \times Y$ is the topology $\tau$ whose basis is given by:

(1)
\begin{align} \quad \mathcal B = \{ U \times V : U \: \mathrm{is \: open \: in \:} X, \: V \mathrm{is \: open \: in \:} Y \} \end{align}

The product $X \times Y$ with the product topology $\tau$ is called the topological product of the spaces $X$ and $Y$.

Similarly, if $X_1, X_2, ..., X_n$ is a finite collection of topological spaces and $\displaystyle{\prod_{i=1}^{n} X_i = X_1 \times X_2 \times ... \times X_n}$ then the product topology on $\displaystyle{\prod_{i=1}^{n} X_i}$ is the topology $\tau$ whose basis is given by:

(2)
\begin{align} \quad \mathcal B = \left \{ \prod_{i=1}^{n} U_i : U_1 \: \mathrm{is \: open \: in \:} X_i, \: \forall i \in \{ 1, 2, ..., n \} \right \} \end{align}

Notice that the order of the topological spaces that constitute the topological product matter to some degree. If we instead have an infinite number of topological spaces and we want to consider the resulting topological product then a clear order in the notation above may not make sense (especially if we're considering an uncountable number of topological spaces). Consequentially we define arbitrary topological products in a different manner.

Let $\{ X_i \}_{i \in I}$ be an arbitrary collection of topological spaces where $I$ is an indexing set, and let $\displaystyle{\prod_{i \in I} X_i}$ be the Cartesian product of these spaces. We can think of the elements in $\displaystyle{\prod_{i \in I} X_i}$ as functions $f : I \to X_i$ defined for all $i \in I$ by $f(i) = x_i$ for $x_i \in X_i$, or unordered "sequences" $(x_i)_{i \in I}$ where $x_i \in X_i$.

 Definition: Let $\{ X_i \}_{i \in I}$ be an arbitrary collection of topological spaces and let $\displaystyle{\prod_{i \in I} X_i}$ be the Cartesian product of these spaces. The Product Topology on $\displaystyle{\prod_{i \in I} X_i}$ is the topology $\tau$ is the initial topology induced by the projection maps $\displaystyle{p_i : \prod_{i \in I} X_i \to X_i}$. The corresponding Topological Product is the topological space $\displaystyle{\left ( \prod_{i \in I} X_i, \tau \right )}$.

Recall that the initial topology induced by a collection of maps $\{ f_i : i \in I \}$ has the following subbasis:

(3)
\begin{align} \quad \mathcal S = \{ f_i^{-1}(U) : U \in \tau_i \} \end{align}

So the initial topology on $\displaystyle{\prod_{i \in I} X_i}$ induced by the collection of projection maps $\{ p_i : i \in I \}$ has the following subbasis:

(4)
\begin{align} \quad \mathcal S = \left \{ U = \prod_{i \in I} U_i : V = p_i^{-1}(U) \: \mathrm{for \: some \: open} \: V \subseteq X_i \: \mathrm{and} \: i \in I \right \} \end{align}

In other words, the elements in the subbasis $\mathcal S$ are the arbitrary products $\displaystyle{U = \prod_{i \in I} U_i}$ such that the inverse image of this product $U$ with respect to some projection map $p_i$ is equal to an open set $V$ in the corresponding topological space $X_i$.