Product Topology

Definition

The product topology on $\Pi X_{\lambda }$, $X_{\lambda }$ topological spaces, is the initial topology induced by the family of projections ${\Pi }_{\lambda }$

What is ${\Pi }_{\lambda }$?

${\pi }_{\lambda }:{\Pi }_{{\lambda }^{\prime }\in \wedge }{X}_{{\lambda }^{\prime }}\to X_{\lambda }$
${\pi }_{\lambda }(f)=f(\lambda )$
${\pi }_{\lambda }((X_{{\lambda }^{\prime }}))=x_{\lambda }$

$\underset{\in (x_{\lambda }{)}_{\lambda \in \wedge }}{\underbrace{{\Pi }_{\lambda \in \wedge }{X}_{\lambda }\equiv }}\{f:\wedge \to {\cup }_{\lambda \in \wedge }{X}_{\lambda }|\underset{x_{\lambda }}{\underbrace{f(\lambda )}}\in X_{\lambda }\}$

Example

Product of 2 topological spaces: $x_1\times x_2$
$=\{(x_1,x_2)|x_i\in X_i\}$
${\pi }_1((x_1,x_2))=x_1$

Transclude of Drawing-2025-02-24-12.47.32.excalidraw