I was studying Birkhoff's transitivity theorem... I did not have problem with the theorem, but there were somethings in the proof of theorem that I could not understand.
For example if $X$ is a topological space and $T:X \to X $ be given, then it is said that if $ x \in X $ then since scalar multiplication is continuous every neighborhood of $x$ contains points from $ X -\left\{x \right\} $ thus $X$ has no isolated points.
It is not easy for me to understand. Could you please help me ?
For "scalar multiplication" to make any sense $\ X\ $ would need to have more structure than a mere topological space. Most likely would be that of a topological vector space. In that case, if $\ U\ $ is any neighbourhood of $\ x\ne0\ $, then the continuity of scalar multiplication implies that there is a positive $\ \epsilon\ $ such that $\ (1+t)x\in U\ $ for every $\ t\ $ such that $\ |t|\le\epsilon\ $. It then follows that $\ (1+t)x\in U\cap\left(X-\{x\}\right)\ $ for $\ 0<|t|\le\epsilon\ $.
Unless $\ X\ $ is trivial, then by choosing some $\ y\ne0\ $ you can use essentially the same argument for $\ x=0\ $.