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I want to try and understand infinite families of sets better. What are some examples of

(1) an infinite family of open sets whose intersection is not open

(2) an infinite family of closed sets whose union is not closed

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2 Answers 2

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One may consider $\mathbb{R}$ with its usual topology.

  1. We have $\displaystyle \bigcap_{n\geq1}\left]-\frac1n,\frac1n \right[=\left\{0\right\}$

  2. We have $\displaystyle \bigcup_{n\geq1}\left[0,1-\frac1n \right]=\left[0,1\right[$.

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Consider the natural numbers $\Bbb N$ and let $n$ denote a natural number. Let $a \in \Bbb R$ be a fixed real number.

For (1): $\{ (a - \frac{1}{n}, a + \frac{1}{n}) \}_{n \in \Bbb N}$ is a countably infinite family of open intervals. Their intersection is not open. Why? What is $\bigcap \limits_{n =1}^{\infty} (a - \frac{1}{n}, a + \frac{1}{n})$?

For (2): $\{ [ 0 + \frac{1}{n + 4}, 1 - \frac{1}{n + 4}] \}_{n = 1}^{\infty}$ is a countably infinite family of closed intervals whose union is not closed. Why? What is $\bigcup \limits_{n = 1}^{\infty} [ 0 + \frac{1}{n + 4}, 1 - \frac{1}{n + 4}]$?

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