To show [0,1] is not homogeneous

Question

How can we show that [0,1] is not homogeneous topological space?

So far what I have thought is as follows:- If it is homogenous then there exists a homeomorphism between every pair of points in it. I think there is no homeomorphism carrying 0 to any point in (0,1)( in particular there is no homeomorphism carrying 0 to 1/2) and so I tried to get a proof by contradiction but I'm stuck.

Note:-[0,1] is equipped with subspace topology inherited from usual topology on R.

Answer 1

As is well known any continuous one-to-one map on $[0,1]$ is strictly monotonic. If $f$ is one such map with $f(0)=\frac 1 2 $ then $f$ cannot be a surjection: if it is strictly increasing then $f(x) \geq \frac 1 2$ for all $x$, so it cannot take values less than $\frac 1 2$. If it is strictly decreasing then $f(x) \leq \frac 1 2$ for all $x$, so it cannot take values greater than $\frac 1 2$.

Answer 2

What happens when $f(0)\in(0,1)$ when $f$ is continuous and surjective? Since $f(a)=0$ and $f(b)=1$ for some $a,b>0$, then by the intermediate value theorem there is $x$ between $a,b$ (and thus $x\neq 0$) such that $f(x)=f(0)$. Therefore $f$ cannot be injective.

In particular any homeomorphism $[0,1]\to[0,1]$ has to map $0$ either to $0$ or $1$.

Answer 3

Assume there exists a homeomorphism $h : [0,1] \to [0,1]$ such that $h(0) = 1/2$. Then $h$ restricts to a homeomorphism $h' : (0,1] = [0,1] \setminus \{0\} \to [0,1] \setminus \{1/2\}$. This is impossible because $(0,1]$ is connected, but $[0,1] \setminus \{1/2\}$ is not.