Munkres order topology difference between definition of simple order using $<$ instead of $\leq$

Question

Munkres in his Topology 2ed section 14 p. 84 defines a simply ordered set as in this math.stack question here

definition of simply ordered set by Munkres

I recalled that I have come across the definition of poset and total order in my discrete math course and also in my algebra course when discussing lattices but I vaguely remembered that the order relation was defined in terms of $\leq$ and not as $<$ like in Munkres. I.e. like on the following wiki page

wiki definition of simple order

Now, my question is. What is the difference between the two definitions and why did Munkres choose $<$ and not $\leq$?

Answer 1

I'm pretty certain Munkres does it the way he does because when you define the order topology on a totally ordered set, you need to use $<$ to define open intervals. I think it is more common to use $\leq$ than $<$ in general when speaking about orders, but in this case, Munkres has specific use for $<$ where $\leq$ just won't work, and therefore it's reasonable to let that be the symbol of choice.

Answer 2

If you choose $\leq$ instead of $<$ then you get that $\uparrow x \; \cap \downarrow x = \{x\}$ is open for $x\in P$ (where $P$ is the poset in question), so you end up with the discrete topology, which is very uninteresting. (We set $\uparrow x \; = \{y \in P: y\geq x\}$, and similarly for $\downarrow x$.)

Answer 3

I think you should ask Munkres.

Don't worry though because every irreflexive partial ordening $(P,<)$ goes "hand in hand" with a reflexive partial ordening $(P,\leq)$.

You can construct the reflexive from the irreflexive by adding diagonal $\Delta=\{(p,p)\mid p\in P\}$ and you can construct the irreflexive from the reflexive by removing the diagonal $\Delta=\{(p,p)\mid p\in P\}$.

I think it is just a matter of taste.

Answer 4

Munkres in his book only wants to consider "linear orders" in the topological context (this is not strictly needed, there is also theory on partially ordered sets and natural topologies on those) and there $<$ is quite natural: we have a natural three-way split (trichotomy):

$$\forall x,y: (x=y) \lor (x < y) \lor (x > y)$$

because all elements are comparible. Open sets (the subbasic ones) are defined by using $<$ and so he wants that as the primary notion. Then he has a derived notion $x \le y$ for $x < y$ or $x=y$ when he doesn't want to exclude equality.

He can still use consistently the symbol $\le$ for any partial order too, as the introduced $\le$ for a linear order also obeys the anti-symmetry property

$$x \le y \land y \le x \to x=y$$ so both symbolic uses obey the same axioms (reflexivity, antisymmetry and transivity), but $<$ in his text will always be unambiguously a linear and strict order.