generated from smyalygames/quartz
842 B
842 B
Definition
The product topology on \Pi X_{\lambda}, X_{\lambda} Topological Space, is the Initial Topology induced by the family of projections \Pi_{\lambda}
[!note] What is
\Pi_{\lambda}?\pi_{\lambda} : \Pi_{\lambda' \in \wedge} X_{\lambda'} \to X_{\lambda}\pi_{\lambda}(f) = f(\lambda)\pi_{\lambda}((X_{\lambda'})) = x_{\lambda}
\underbrace{\Pi_{\lambda \in \wedge} X_{\lambda} \equiv}_{\in (x_{\lambda})_{\lambda \in \wedge}} \{ f : \wedge \to \cup_{\lambda \in \wedge} X_{\lambda} \; | \; \underbrace{f(\lambda)}_{x_{\lambda}} \in X_{\lambda} \}
Example
Product of 2 Topological Space: x_{1} \times x_{2}
= \{ (x_{1}, x_{2}) \; | \; x_{i} \in X_{i} \}
\pi_{1}((x_{1}, x_{2})) = x_{1}
!Drawing 2025-02-24 12.47.32.excalidraw