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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