Let $X$ be a compact Hausdorff space and let $X_1 \subset X_2 \subset X_3 \subset \cdots$ be a sequence of closed, connected subspaces. Prove that $\bigcap_{i=1}^\infty X_i$ is connected. Give an example showing that the compactness of $X$ is necessary
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2:what are you trying ? – M.H Jun 20 '13 at 20:28
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http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2005;task=show_msg;msg=1889.0001 – Henno Brandsma Jun 20 '13 at 20:32
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7If $X_1 \subset X_2 \subset \cdots$, then $\bigcap_i X_i = X_1$. Do you mean $X_1 \supset X_2 \supset \cdots$? – Ayman Hourieh Jun 20 '13 at 20:37
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see also: http://math.stackexchange.com/q/383841/49437 – Martin Jun 20 '13 at 21:20
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Edit: As mentionned by Ayman Hourieh, I supposed $X_1 \supset X_2 \supset \dots$ in my answer.
Suppose that $\bigcap\limits_{i \geq 1} X_i$ is not connected.
- Show that there exist two disjoint open sets (in $X$) $U$ and $V$ such that $\bigcap\limits_{i \geq 1} X_i \subset U \cup V$.
- Show that But $F= X \backslash (U \cup V)$ is closed and $X_i \cap F \neq \emptyset$ for all $i \geq 0$.
- Deduce that $\bigcap\limits_{i \geq 1} (X_i \cap F)$ is nonempty.
- Find a contradiction.
For a counterexample, try $X_n= ([2-1/n,2+1/n] \times \mathbb{R}) \backslash \left( (-1,1) \times (-n,n) \right)$.
Seirios
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