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Is the union of any collection of closed sets always closed?

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Q: Is the union of any collection of closed sets always closed?

A: No.

Why: Try this

$$\left[0, 1 - \frac{1}{n} \right], n = 1, 2, 3, ..,$$

  • Can such a union be open?
  • Under what circumstances is such a union closed? Answer.
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