Defining an upper/lower bound in lexicographically ordered C.

[c0dy]

If I have a lexicographic ordering on ℂ, and I define a subset, $A = \{z \in ℂ: z = a+bi; a,b \in ℝ, a<0\}$.

I have an upper bound, say α = 0+di. My question is does only the real part, Re(α) = 0 define the upper bound? Or does the Im(α) = d have nothing to do with bounds in general?

Since it seems to me if I have the lexicographic ordering on ℂ such as for any two m,n $\in$ ℂ, where m = a+bi and n = c+di and I define the ordering as m<n if a<c or if a=c and b<d.

The last bit, b<d gives me the impression that Im(α) would play a role in the upper bound. The reason I am asking is because in a proof I read, they prove this order has no least upper bound as there are infinitely many complex numbers with their real parts equal to Re(α) but different imaginary parts. So, I guess if only the real parts of complex numbers define the bounds then it makes sense to me.

 

[SteveL27]

If I have a lexicographic ordering on ℂ, and I define a subset, $A = \{z \in ℂ: z = a+bi; a,b \in ℝ, a<0\}$.

I have an upper bound, say α = 0+di. My question is does only the real part, Re(α) = 0 define the upper bound? Or does the Im(α) = d have nothing to do with bounds in general?

Since it seems to me if I have the lexicographic ordering on ℂ such as for any two m,n $\in$ ℂ, where m = a+bi and n = c+di and I define the ordering as m<n if a<c or if a=c and b<d.

The last bit, b<d gives me the impression that Im(α) would play a role in the upper bound. The reason I am asking is because in a proof I read, they prove this order has no least upper bound as there are infinitely many complex numbers with their real parts equal to Re(α) but different imaginary parts. So, I guess if only the real parts of complex numbers define the bounds then it makes sense to me.

A least upper bound has to be a specific number with the LUB property. In this case there is no such number, since there are lots of upper bounds but none of them is the smallest.