An example illustrating that an infinite union of closed sets is not always closed PDF

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An example illustrating that an infinite union of closed sets is not always closed.

Let Sn ≡ [(1/n)- 1, 1-(1/n)] for n = 1,2,3,……. Note that each Si is a closed set for i = 1,2,… So S1 = {0} [S1 is the single element set containing only 0 which is a closed set] S2 = [- ½ , ½]. But note that S1 is a subset of S2 and hence S1 U S2 = S2 Similarly, S2 U S3 = S3 = [-2/3, 2/3]

In general, Sn U Sn+1 = Sn+1 Note also that as n tends to infinity Sn converges to the set (-1, +1). This is because the left point of the interval of Sn will never equal -1 and always be a little more than -1. Similarly, the right point will never equal +1 but always be a little less.

Hence ⋃∞ 𝒏=𝟏 𝑺𝒏 = (−𝟏, +𝟏) which is not a closed set....