Understanding Clopen Sets in X: A Wikipedia Example

[theneedtoknow]

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

 

[matt grime]

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]

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.

 

[theneedtoknow]

Ah ok, thanks guys :) its more clear now