I've been confused about the definition of **directed set** in John L. Kelley's book "General Topology" (1), and feel inconsistency in multiple relevant places.

Overall, I feel that in the following sentence on [p.65](https://i.stack.imgur.com/TKms6.png)

> 
> A binary relation $\geq$ **directs** a set *D* if *D* is non-void and ...
> 
> 
> 

the symbol "$\geq$" should have been "$\leq$". Such change would make it consistent with other textbooks, such as Munkres (2) and Willard (3).

More importantly, Kelley seems to be inconsistent with himself, e.g.,

1. in the paragraph right below the definition of **directs** on [p.65](https://i.stack.imgur.com/TKms6.png), it says

> 
> We say that *m* follows *n* in the order "$\geq$" and ... iff $m \geq n$
> 
> 
> 

while earlier on [p.13](https://i.stack.imgur.com/eUKvG.png) he said

> 
> If $<$ is an ordering and $x<y$, then ... $y$ follows $x$
> 
> 
> 

which seems more natural to me.

The inconsistency between the two places is that if we treat $<$ and $\geq$ as a more abstract relation $R$, then on [p.65](https://i.stack.imgur.com/TKms6.png) he says $m$ follows $n$ if $\mathbf{mRn}$, while on [p.13](https://i.stack.imgur.com/eUKvG.png) he says $m$ follows $n$ if $\mathbf{nRm}$ (I made substitution $x\rightarrow n$ and $y\rightarrow m$ to make the contrast more clear).

2. In the paragraph that follows, there is another example:

> 
> ... the set $\omega$ of non-negative integers are directed by $\geq$. Observe that 0 is a member of $\omega$ which follows every other member in the order $\leq$.
> 
> 
> 

From the context I feel that the $\geq$ in the above text should be interpreted as the usual ordering of natural numbers, i.e., $j \geq i$ means $j$ is bigger than or equal to $i$ in the usual sense. If we follow this interpretation, then I do not see how the property of 0 explains why $\omega$ is directed by $\geq$. In contrast, the property of 0 would explain why $\omega$ is directed by $\leq$.

I've also tried hard with alternative interpretations in multiple places, but could not make Kelley's exposition consistent with itself. Anybody has a clue?

(1) Kelley, *General Topology*, Dover edition 0-486-81544-7   

(2) Munkres, *Topology: a first course*, Prentice-Hall 1975, 0-13-925495-1   

(3) Willard, *General Topology*, Dover edition, 0-486-43479-6   

Attached are the pictures of [p.13](https://i.stack.imgur.com/eUKvG.png) and [p.65](https://i.stack.imgur.com/TKms6.png) from the Springer edition of GTM27 0-387-90125-6, which is the same as (1).