## 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 ?

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Look for Bourbaki's definition of "ordered". – Gerald Edgar Jan 30 2012 at 14:05
I looked it up, and "order" does, indeed, mean partial order. – Gerald Edgar Jan 30 2012 at 14:13
errata = a list of errors in a published work. So it's not an "errata", but possibly (at most) an error. But I don't think it is, it's correct as stated. – Henno Brandsma Jan 30 2012 at 14:27
erratum is the singular form. :) – Jim Conant Jan 30 2012 at 15:55
Still I would also suggest to avoid using erratum in this form, as it is my impression its standard meaning in academic writing is a bit different from the pure translation 'error'; often referring to some (informally) published note pointing out and possibly fixing an error. – quid Jan 30 2012 at 17:50

## 1 Answer

Say we have a partially ordered set. What so 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...

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 I solved this Exercise using the idea of Poset only, but then I got confused after reading the definition of right and left topology in Wiki: en.wikipedia.org/wiki/…. In General Topology -I, 1.4, while giving definition of Closed Set, Bourbaki says: On the rational line, every interval of the form [a,→[ is a closed set, for its complement ]←,a[ is open. But I agree that closed/open sets are subject to Topology under consideration. – Reetesh Mukul Jan 30 2012 at 14:45