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union-of-connected-subsets-is-connected-if-intersection-is-nonempty [duplicate]

Union of connected subsets is connected if intersection is nonempty I don't understand why A∩F and B∩F are relatively open where Brian Scott commented. Thanks
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Show that if the intersection is non-empty, then the union is connected of the following [duplicate]

Let $\{A_i\}_{i \in I}$ be a family of connected subsets of a metric space $X$ ($I$ is some set of indices). Show that if the intersection $\bigcap A_i \neq \emptyset$ , then $\bigcup A_i$ is ...
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Proving that $\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$ is connected

Let $E=\{(x,y):x^2-2x+y^2=0\}\cup \{(x,0):x\in [2,3]\}$. It appears to me from the sketch that this set is connected. However, I have no idea how to prove this, since the set is not convex, and we ...
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Proving a set is connected in $\mathbb{R}^2$

Let $B = \{(x,y) \in \mathbb{R}^2 : x\in \mathbb{Q} \text{ or } y\in \mathbb{Q}\}$. Prove that $B$ is either connected or disconnected. I am not sure if this logic is correct but I believe its ...
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if union of connected subsets are connected in a metric space, what if the union is the metric space $X$ itself?

With reference to this link that i read: Union of connected subsets is connected if intersection is nonempty My question is: if union of connected subsets are connected in a metric space and their ...
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$\textbf{Question:}$ If $\{E_{\alpha} \}_{\alpha \in A}$ is a collection of connected subsets of $X$ such that $\bigcap_{\alpha \in A} E_{\alpha} \neq \emptyset$, then $\bigcup_{\alpha \in A} E_{\... 2answers 68 views $\mathbb{R}^{p}\backslash \{0 \}$is connected for$p \geq 2$I have to show that$\mathbb{R}^{p}\backslash \{0 \}$is connected for$p \geq 2$. Is it possible to show this using the property debated on in this article: Union of connected subsets is connected if ... 2answers 95 views Elementary topology of$\mathbb C\$: Union of 2 regions with nonempty intersection is a region

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 1.31,32 Perhaps related but my topology is so far limited to the elementary topology in ...