Examples of ordered topology on R x R

[Pippi]

I am trying to understand the difference between ordered topology and subspace topology. For one, how do I write down ordered topology of the form {1} x (1, 2] ? How do I write down a basis for {1,2} x Z_+ ?

 

[micromass]

What order are you planning to put on these spaces?? (specifically the second one)

 

[Pippi]

I want to put dictionary order on them, so (1, 2) is less than (2, 2) and also less than (1,3). I don't know other orders can be put on them (maybe a metric?) than dictionary order.

 

[micromass]

So, the order topology is by definition generated by intervals. That is

$$\{x~\vert~a<x<b\}$$

Can you give a description of the intervals in your two spaces?

 

[Pippi]

For the space {1} x (1, 2], the intervals are half open intervals {x | (1, 1] < (1, x] < (1, 2]}. I can also define open intervals {x | (1, 1) < (1, x) < (1, 2)} and a half open interval about {1, (1, 2]}. Both are valid topology.

The space {1,2} x Z_+ contains the set {{1, 1} {1, 2} ... {2, 1}, {2, 2} ... }. Because I can define the basis {x | (x - 1, x + 1), x in Z_+} for space Z_+, the basis are {{1, (x - 1, x+1), {2, (x - 1, x +1)}}. The basis for x = 1 are {1, [1, 2)} and {2, [1, 2)}.

I can't find errors but this is suggesting that I can find many different order topology on a set? Infinitely many? :/.