Product Space

Universal properties of Product space

let ${\Pi }_A$ and ${\Pi }_B$ be projection maps.

Let $T$ be a arbitrary space that has maps $t_A$ and $t_B$ to $A$ and $B$, then there is a unique continuous map from $T$ to $A\times B$ defined by $\mathcal{U}:=(t_A(x),t_B(x))$ to make left and right commutes (universal property of product in the category of topological spaces) That is ${\Pi }_A\circ \mathcal{U}=t_A$ and ${\Pi }_B\circ \mathcal{U}=t_B$ In fact, any set that satisfies this property (has a map that makes left and right commute), then it has to be isomorphic to the definition of $A\times B$

we want the projection function on the topology $A\times B$ to be continuous

Definition

Let $X_1,\dots ,X_n$ be spaces. The product topology on $X_1\times \cdots \times X_n$ is generated by the basis

$$\mathcal{B}=\{{\mathcal{U}}_1\times \cdots \times {\mathcal{U}}_n|{\mathcal{U}}_i\text{ open in }{X}_i\forall i\}$$

function family

given continuous maps $f_i:X_i\to Y_i$ define $f_1\times \cdots \times f_n:X_1\times \cdots \times X_n\to Y_1\times \cdots \times Y_n$

$$(x_1,\dots ,x_n)\mapsto ,(f_1(x_1),\dots ,f_n(x_n))$$

a general indexed family of space can be notated as ${\Pi }_{\alpha \in A}{X}_{\alpha }$

Characteristic Property

For any space $Y$ and function $f:Y\to {\prod }_{\alpha \in A}{X}_{\alpha }$ $f$ is continuous $⟺$ $f_{\alpha }={\Pi }_{\alpha }\circ f$ is continuous for all $\alpha \in A$

Warning

Note that $\Pi$ \Pi is different from $\prod$ \prod

We try to prove the $⟸$ direction

Proof using universal property

There exists a unique continuous map $\mathcal{U}:Y\to {\prod }_{\alpha \in A}{X}_{\alpha }$ that makes the graph commute This gives ${\Pi }_i\circ \mathcal{U}=f_{\alpha }$ for all $\alpha \in A$ By assumption, ${\Pi }_{\alpha }\circ f=f_a$ for all $\alpha \in A$

Consider the underlying sets of these spaces, $⟹f=\mathcal{U}$ as function Since $\mathcal{U}$ is continuous $⟹$ $f$ is continuous

Properties:

• If each $f_i$ is continuous, so is $f_1\times \cdots \times f_n$
• If each $f_i$ is a homeomorphism, so is $f_1\times \cdots \times f_n$ Product space preserves:
• Hausdorff property: If every $X_{\alpha }\in A$ is Hausdorff, then ${\prod }_{\alpha \in A}{X}_{\alpha }$ is Hausdorff
• First countability (over countable $A$)
• Second countability (over countable $A$)
• Connectedness and Path Connectedness (finitely many)