i'm having a problem with the definition of open, closed and clopen sets.

I have understood the basic definitions, but then the teacher today in class said that the normed space is limitless. A colleague of mine said then that the normed space is clopen, but i can't really see why. To me, a normed space, since it has no limits, it is impossible to find a ball that has points not belonging to the normed space, so it can't be a closed set, right? Then, it must be an open set. Or am I mixing everything up? Thanks in advance!

  • $\begingroup$ Any metric space, normed or otherwise, is both closed and open as a subset of itself. A normed space may, however, be a closed subset of a larger normed space with the inherited norm. $\endgroup$ – Omnomnomnom Feb 6 '15 at 16:39

For any topological space $(X,\tau)$, $X$ itself (and the empty set, btw) is always both open and closed (clopen, for short). So if you look at your normed space as a topological space, it is both.

(if you are looking at a normed space $X$ which is a subspace of another space $Y$, $X$ need not be closed in $Y$)


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