Why is it that a metric space (X,d) always has two clopen subsets; namely {0}, and X itself?(adsbygoogle = window.adsbygoogle || []).push({});

Rudin calls it trivial, and so do about 15 other resources I've perused.

What confuses me is that if we define some metric space to be the circle in ℝ: x^{2}^{2}+y^{2}≤ r^{2}, then points on the boundary of the circle don't have neighborhoods contained entirely in X, since for any radius > 0, the neighborhood will extend out of X.

Thanks!

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# Metric spaces - Clopen property

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