I must be overlooking something! Given a metric space (E,d), the improper subset E is open in E. How? Here is my understanding:(adsbygoogle = window.adsbygoogle || []).push({});

1) We call a set S(subset of E) open iff for all x(element of S) there exist epsilon such that an open ball of radi epsilon centered about s is wholly contained in S.

So, how can E always be open. Here is my counterexample:

Take E to be a subset of R^2 described by...

E={(x,y):0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 }

If now analyze E as a subset of itself, the set E is clearly not open as it contains its boundary. What am I missing here?

Thanks,

Topology Newbie

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# Given a metric space (X,d), the set X is open in X. HELP!

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