Prove that the closed interval [0,1] is not a homogeneous topology by showing that there's no bijective, open and continuous (bi-continuous) mapping h: [0,1]→[0,1] such that h(1/2)=0.
The closed interval is equipped with the usually metric. If the mapping mentioned above exists for any x, y such that h(x)=y, then the topology is called homogeneous.
The Attempt at a Solution
The book says it is easy to verify this; I can kind of feel that it doesn't exist, since if h(1/2)=0 then h(1/2+)=h(1/2-) which violates the bijection, but this is obviously not a proof. Any suggestions?