Confusion with Disconnected sets

[Mr-T]

Hello,

I am having some difficulties understanding why a subset under the usual metric topology of the reals is connected.

How can a set X = (0,1] u (1,2) be connected?

The definition I am using is:

A is disconnected if there exists two open sets G and V and the following three properties hold:

(1) A intersect G ≠ ∅
A intersect V ≠ ∅

(2) A is a proper subset of the union of G and V.

(3) the intersection of G and V is the empty set.

Thanks

 

[micromass]

And you also want $G$ and $V$ to be open. No?

 

[Mr-T]

Yea, that would be more correct.

 

[HallsofIvy]

Okay, so can you find two such open sets for (0, 1]\cup (1, 2)? (0, 1] and (1, 2) will not do because (0, 1] is not open. (And, did you notice that (0, 1]\cup (1, 2)= (0, 2)?)