Finite Space that is Not Normal

Question

Is there any finite space that is not normal? By "normal", I refer to a space in which disjoint closed sets can be separated by disjoint open sets.

Answer 1

You can take $\{a,b,c\}$ with open sets $\emptyset$, $\{b,c\}$, $\{a,c\}$, $\{c\}$, $\{a,b,c\}$.

Then $\{a\}, \{b\}$ are disjoint closed sets. The open sets that contain them, $\{a,c\}$, $\{b,c\}$, $\{a,b,c\}$ are never disjoint.

Answer 2

One of the other answers is a special case of the finite particular point topology, which is defined as follows:

Let $X$ be any finite set and fix a particular point $p$ of $X$. A set is open in this topology if it contains $p$ or is empty.

Now, any two nonempty open sets must intersect in at least the point $p$, so there are no nonempty, disjoint open sets. In particular, the space is not normal.