Point Metric
How TO PROVE that a SUBSET of METRIC SPACE is CLOSED if and only if it contains all its CLUSTER POINTS?
Please write in your own words, so I can understand it, I have found proofs online, which I cannot follow…
Thanks for your help
you don’t even need it to be a metric space.
Let T be any topological space, and take S an arbitrary set of T. S is closed if and only if S is it’s own closure. As the closure of S is S union the limit points of S, S is it’s own closure if and only if S=S union S’, where S’ is the limit points of S. S=S union S’ iff S’ is a subset of S. S’ is a subset of S iff S contains all of it’s limit points.
Adland by James P. Othmer
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