So I've just started trying to teach myself some topology and I've been reading through topology by munkres. In the first chapter he defines order relations being a binary relation satisfying transitivity, nonreflexivity and comparability this has caused me a lot of confusion and I am struggling to grasp the concepts.
My main confusion is to do with that of comparability what exactly is it on Wikipedia the definition of a total order (which is what munkres is talking about in his book) is a binary relation satisfying transitivity,antisymmetry and connexity.
my question is what is connexity is it the same as comparability and how are these 2 definitions equivalent?.
Thanks in advance.
The difference in definitions is that Wikipedia defines non-strict total orders (things like $x\le y$ where elements relate to themselves) and Munkres defines strict total orders (things like $x<y$ where elements do not relate to themselves). Wikipedia's connexity condition states that for any $x,y$ either $x\le y$ or $y\le x$. This is not true for strict total orders in the case where $x=y$, so Munkres' definition instead states for any distinct $x,y$ either $x<y$ or $y<x$. Annoyingly "total order" as used in math is ambiguous, we precede it by "strict" or "non-strict" if we want to distinguish which we are talking about.
For what it's worth "connex" is quite rare, I don't think I've ran into it outside Wikipedia. According to this talk page it was introduced because the previous language (calling this a "total" relation) was ambiguous.