esperant wrote:Given any two distinct points in X (say 0.5 and 0.7) there is are disjoint open neighbourhoods of the two points (e.g. [.49,.51) and [.69,.71). Therefore II.
Consider the open cover of X given by [0,0.9), [0.9,0.99), [0.99, 0.999), etc and [1,2). This is obviously doesn't have a finite subcover. Therefore not I.
The cover from the above paragraph was disjoint, so X is not connected. Therefore not III.