Finite Topological Products Review

We will now review some of the recent material regarding topological products.

• On the Finite Topological Products of Topological Spaces page we said that if $X$ and $Y$ are topological spaces then the resulting Topological Product is the set $X \times Y$ with the topology $\tau$ called the Product Topology on $X \times Y$ 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}

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

(2)

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

• We also saw that the topological products $X \times Y$ and $Y \times X$ are homeomorphic to each other. An explicit homeomorphism between these two spaces is given for all $(x, y) \in X \times Y$ by:

(3)

\begin{align} \quad f(x, y) = (y, x) \end{align}

• On the Projection Mappings of Finite Topological Products page we said that if $\{ X_1, X_2, ..., X_n \}$ is a finite collection of topological spaces and $\displaystyle{\prod_{i=1}^{n} X_i}$ is the corresponding topological product then for each $j \in \{ 1, 2, ..., n \}$ the corresponding projection maps $\displaystyle{p_j : \prod_{i=1}^{n} X_i \to X_j}$ are defined for all $\mathbf{x} = (x_1, x_2, ..., x_n)$ by:

(4)

\begin{align} \quad p_j(\mathbf{x}) = p_j(x_1, x_2, ..., x_n) = x_j \end{align}

• We saw that each of the projection maps $p_j$ are surjective, open, and continuous. Furthermore, we saw that the product topology on $\displaystyle{\prod_{i=1}^{n} X_i}$ is simply the coarsest topology which makes the maps $\{ p_1, p_2, ..., p_n \}$ continuous, i.e., the product topology is the initial topology induced by $\{ p_1, p_2, ..., p_n \}$.

• We then looked at a ton of nice properties of topological products. If $\{ X_1, X_2, ..., X_n \}$ is a finite collection of topological spaces then:

Page	Property	Consequence
The Open and Closed Sets of Finite Topological Products	a) $U_i \in X_i$ is open for all $i \in \{ 1, 2, ..., n \}$ b) $C_i \in X_i$ is open for all $i \in \{ 1, 2, ..., n \}$	a) $\displaystyle{\prod_{i=1}^{n} U_i}$ is open in $\displaystyle{\prod_{i=1}^{n} X_i}$. b) $\displaystyle{\prod_{i=1}^{n} C_i}$ is closed in $\displaystyle{\prod_{i=1}^{n} X_i}$.
The Interior of Sets in Finite Topological Products	$A_i \subseteq X_i$ for all $i \in \{ 1, 2, ..., n \}$.	$\displaystyle{\mathrm{int} \left ( \prod_{i=1}^{n} A_i \right ) = \prod_{i=1}^{n} \mathrm{int} (A_i)}$.
The Closure of Sets in Finite Topological Products	$A_i \subseteq X_i$ for all $i \in \{ 1, 2, ..., n \}$.	$\displaystyle{\overline{\left ( \prod_{i=1}^{n} A_i \right )} = \prod_{i=1}^{n} \overline{A_i}}$.
The Set of Accumulation Points in Finite Topological Products	$A_i \subseteq X_i$ for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\left ( \prod_{i=1}^{n} A_i \right )' \supseteq \prod_{i=1}^{n} A_i'}$.
Dense Sets in Finite Topological Products	$A_i \subseteq X_i$ is dense in $X_i$ for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} A_i}$ is dense in $\displaystyle{\prod_{i=1}^{n} X_i}$.
First Countability of Finite Topological Products	$X_i$ is first countable for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} X_i}$ is first countable.
Second Countability of Finite Topological Products	$X_i$ is second countable for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} X_i}$ is second countable.
Separability of Finite Topological Products	$X_i$ is separable for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} X_i}$ is separable.
The Hausdorff Property on Finite Topological Products	$X_i$ is Hausdorff for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} X_i}$ is Hausdorff.
Metrizability of Finite Topological Products	$X_i$ is metrizable for all $i \in \{1, 2, ..., n \}$.	$\displaystyle{\prod_{i=1}^{n} X_i}$ is metrizable.