Quartz sync: Mar 1, 2025, 2:26 PM

content/Definitions/Topological Spaces/Induced/Initial Topology.md

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+# Definition
+The **initial topology** on $X$ induced by a family of functions $f : X \to Y_{f}$ into [[Topological Space|topological spaces]] $Y_{f}$ is the [[Weakest Topology|weakest topology]]on $X$ making all these functions [[Continuous|continuous]].
+
+Here: $F = \{ f^{-1} \; | \; f : X \to Y_{f}, \; A \; \text{open in} \; Y_{f} \}$.

content/Definitions/Topological Spaces/Induced/Product Topology.md

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+# Definition
+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 $\Pi_{\lambda}$
+> [!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|topological spaces]]: $x_{1} \times x_{2}$
+$= \{ (x_{1}, x_{2}) \; | \; x_{i} \in X_{i} \}$
+$\pi_{1}((x_{1}, x_{2})) = x_{1}$
+![[Drawing 2025-02-24 12.47.32.excalidraw]]

content/Definitions/Topological Spaces/Induced/Separating Points.md

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+# Definition
+A family $F$ of functions on a set **separating points** $x \neq y$ in the set if $f(x) \neq f(y)$ for some $f \in F$

content/Definitions/Topological Spaces/Induced/Weakest Topology.md

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+# Definition
+Given $F \subset \wp(X)$. The **weakest topology** on $X$ that contains $F$ is the intersection of all the [[Topology|topologies]] that contains $F$. This is a [[Topology|topology]], and consists of $\emptyset$, $X$, and all unions of finite intersections of members from $F$.
+
+> [!example]
+> $F \subset \tau$
+> $\textvisiblespace \cap \tau \ni U_{i} \implies U_{i} \in \tau$
+> $\implies \cap_{i \in F} \; U_{i} \in \tau \implies \cap U_{i} \in \cap_{F \in \tau} \;\tau$
+> 
+> $x_{i} \to x$
+> $\exists j$ such that $x_{i} = x, \; i \geq j$.
+