Exercise 1, § 8

a) Let $X$ be a topological space. Show that the following three statements are equivalent:

(Q) If $x, y$ are any two distinct points of $X$, there is a neighbourhood of $x$ which does not contain $y$.

(Q') Every subset of $X$ which consists of a single point is closed in $X$.

(Q'') For every $x \in X$ the intersection of the neighbourhoods of $x$ consists of the point $x$ alone.

A space $X$ is said to be accessible if it satisfies these conditions.

b) An ordered set $X$ with the right topology (with respect to which $X$ is a Kolmogoroff space [§ 1, Exercise 2]) is accessible if and only if no two distinct elements of $X$ are comparable, and then the right topology on $X$ is the discrete topology.