Finite Hausdorff space is totally disconnected

Question

A space is said to be totally disconnected if its only connected subsets are one point sets.

To show that finite Hausdorff space is totally disconnected.

Can anyone give some hints.

Answer 1

Hint: Any point of a Hausdorff space forms a closed subset.

Answer 2

Suppose a finite Hausdorff space $X$ had a connected subset $C$ with $n>1$ distinct points, say $x_1, \cdots, x_n$. Being Hausdorff, there exists disjoint open sets $U_1$ and $V_2$ in $X$ such that $x_1\in U_1$ and $x_2\in V_2$. Similarly there exists disjoint open sets $U_2$ and $V_3$ such that $x_1\in U_2$ and $x_3\in V_3$ and so on there exists disjoint open sets $U_{n-1}$ and $V_n$ such that $x_1\in U_{n-1}$ and $x_n\in V_n$. Then $\cup_{i=2}^n(C\cap V_i)$ and $\cap_{i=1}^{n-1} (C\cap U_i)$ forms a separation of $C$ which is a contradiction.

Answer 3

A finite Hausdorff space is discrete and all discrete spaces are zero dimensional.