I've been recently trying to teach myself some topology and in the book I'm reading there is the definition of an order relation which confuses me a lot
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I've looked online for more information and they seem to define an order relation differently namely one that is transitive,antisymmetric and reflexive.
So why does the author define it in this way my 2nd confusion has to do with what comparability actually means as there is actually no definition online and I cant quite understand what a connex relation is either and that seems to come up a lot in the definition of an order relation.
Thanks in advance.
Best Answer
Traditonally, we first define a partial order (sometimes even a preorder before that) to be reflexive, antisymmmetric ($xRy$ and $yRx$ implies $x=y$, this a pre-order need not be) and transitive. Lots of examples of this: order on $\Bbb Q$ adn $\Bbb Z$, inclusion relations etc. etc.
Then among those we single out linear orders (the orders we grow up with on "numbers" are mostly linear): we want comparability: $x=y$ or otherwise $xRy$ or $yRx$ must hold (not both as this would imply $x=y$ again). But then peope wanted to axiomatise $x < y$ instead of $x \le y$ and the distinguishing things there are: no reflexivity (never $x< x$ so $xRx$ is always forbidden) and asymmetry: if $xRy$ then never $yRx$), so the relation always has to decide on one of them.
A connex relation is sort of a generalisation of the linear order: we always want either $xRy$ or $yRx$. We then don't demand transivity or asymmetry or antisymmetry necessarily. So it can be a very different thing. E.g. After a round-robin tournament in a game without draws, the players will have constructed a connex relation at the end ($xRy$ : $x$ has beaten $y$), which is the also nonreflexive as you cannot beat yourself.
Linear orders I can see the importance of, and they have a rich theory (order types etc) but connex relations don't add much in my opinion. I've never seen them in any of the papers I studied.