Product Spaces

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 $U:=(t_A(x),t_B(x))$ to make left and right commutes. That is ${\Pi }_A\circ U=t_A$ and ${\Pi }_B\circ U=t_B$ 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

$$\mathcal{B}=\{U_1\times \cdots \times U_n|U_i\text{ open in }{X}_i\forall i\}$$

function family

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

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

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 $f_1\times \cdots \times f_n$