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