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Let X be a topological space, and assume that the subsets $A_α$ of $X$ are connected, where $\alpha\in I$, $I$ some index set. Assume there is a connected subset $B\subset X$ such that $A_α\cap B=\emptyset$ for all $\alpha$.

Prove that the union of $B$ and $\bigcup_{\alpha\in I}A_α$ is connected in $X$. In particular, prove that if $\bigcap_{\alpha\in I}A_α \neq \emptyset$, then $\bigcup_{α\in I}A_α$ is connected.

I started by taking some $G = \bigcup_{\alpha\in A}A_α \cup B$ and $U, W \subset X$. From there assume $G = U \cup W$ is a relatively open partition. Then say $B \subset U$. $A_α \cap B \neq \emptyset$ for any $α$ $\implies$ $A_α \cap U \neq \emptyset$ for any $α$ $\implies$ $G \subset U \implies W = \emptyset \implies G$ is connected.

I also have $tA + (1-t)B = f(t)$ where $f:[0,1] \to \mathbb R^n$ to prove $f(t)$ is continuous. Then set $S = \bigcup_{p\in S} I_p =$ connected and I need to prove $\bigcap_{pϵS} I_p \supset {A} \neq \emptyset$.

Now I have been trying to do this problem using all of this, but can't seem to piece it all together. Could someone help me put it all together or let me know what I'm either doing wrong or missing? Thank you.

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What in the world is going on with your notation? In particular, what's the Mars symbol supposed to mean? – Cameron Buie Mar 25 '13 at 4:05
I think I fixed the LaTeX, but I have no idea what your penultimate paragraph is talkinga bout. – Ian Coley Mar 25 '13 at 4:13
I did the paragraph with the G in it and my professor was writing the penultimate paragraph while he was explaining to me what I should be looking for. He sometimes jumps around topics so I don't exactly get that part. The part above is more of what I am concerned with. – D. Truax Mar 25 '13 at 4:21

Your argument showing that $B\cup\bigcup_{\alpha\in I}A_\alpha$ is connected seems fine. To prove the In particular claim, just fix a point $p\in\bigcap_{\alpha\in I}A_\alpha$ and let $B=\{p\}$; then $B\cup\bigcup_{\alpha\in I}A_\alpha=\bigcup_{\alpha\in I}A_\alpha$.

The penultimate paragraph is incomprehensible.

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