Possible errata in Nicolas Bourbaki's General Topology -I, Chapter 1 Exercise 2 ? Here is the text of Exercise:
2 a) Let $X$ be an ordered set. Show that the set of intervals
$\left[x, \rightarrow\right[$       (resp. $\left]\leftarrow, x\right]$)
is a base of topology on $X$; this topology is called the right (resp. left) topology of $X$. In the right topology, any intersection of open sets is an open set, and the closure of $\{x\}$ is the interval $\left]\leftarrow, x\right]  $. 

The above one was from English edition. I translated French edition and found the same text.

Should not be $X$ a totally ordered set ? And is not that the set of intervals should be  $\left]x, \rightarrow\right[$   in place of $\left[x, \rightarrow\right[$ ?
Is this an errata ?
 A: Bourbaki was right :-)   On the other hand, let   $(X\ \le)$   be a partially ordered set. In general the family
$$\mathbf B\ \ :=\ \ \{\ ]x,\rightarrow[\ :\ x\in X\ \}$$
is NOT a topological base for any topology in   $X$.   One reason is trivial: no minimal element belongs to any member of   $B$;   thus if there is any minimal element then   $X$   would not be open.
OK, one could define:
$$\mathbf B\ \ :=\ \ \{X\}\ \cup\ \{\ ]x,\rightarrow[\ :\ x\in X\ \}$$
It will not help. Indeed, here is a characterization of a topological base:
THEOREM   A family   $\mathbf B$   of subsets of   $X$   is a topological base for a topology in   $X\quad\Leftrightarrow$   the following two conditions hold:


*

*  $\bigcup \mathbf B\ =\ X$

*  $\forall_{G\ H\in\mathbf B}\quad G\cap H\ =\ \bigcup\ \{K\in \mathbf B : K\subseteq G\cap H\}  $
Now consider a 5-element set
$$X := \{b\ \ d\ \ A\ \ C\ \ E\}$$
where by definition there are exactly four sharp inequalities   $b < A$   &   $b < C$   &   $d < C$   &   $d < E$.   Then the intersection
$$ ]b,\rightarrow[\ \ \cap\ \ ]d,\rightarrow[\quad=\quad \{ C \} $$
is not a union of any family of open rays   $ ]x,\rightarrow[ $.
A: Say we have a partially ordered set.  What do you doubt?  (1) The set of intervals $\left[x,\rightarrow\right[$ is a base for a topology.  (2) Any intersection of open sets is open.  (3) The closure of $\{x\}$ is $\left]\leftarrow,x\right]$.  They all look OK to me...
