# Clopen sets

This isn't really a homework question, can someone just explain this bit from wikipedia?

consider the space X which consists of the union of the two intervals [0,1] and [2,3]. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set [0,1] is clopen, as is the set [2,3]. This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.

and later:

A set is clopen if and only if its boundary is empty.

Ok...so take the set [0,1] C X where X = [o,1]U[2,3]....how is the boundary of [0,1] empty? Isn't the boundary of [0,1] the 2 points 0 and 1? So I don't really get how [0,1] is clopen in this case

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org
matt grime
Science Advisor
Homework Helper
What is the definition of boundary? Remember this is in the subspace topology and you shouldn't just think that your intuition about [0,1] being a subset of R is correct - after all [0,1] is open and closed...

quasar987
Science Advisor
Homework Helper
Gold Member
Is 0 really in the boundary of [0,1]? By definition, it is so if every open set U of X containing 0 contains points of [0,1] and of X\[0,1]=[2,3]. Well, take for instance the open set (-1,1)nX=[0,1). It does not contain points of [2,3], so 0 is not in the boundary of X.

What happens here is that [0,1] has boundary {0,1} in R, but not in X.

Ah ok, thanks guys :) its more clear now