If I have a lexicographic ordering on ℂ, and I define a subset, [itex]A = \{z \in ℂ: z = a+bi; a,b \in ℝ, a<0\}[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

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 [itex]\in[/itex] 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Defining an upper/lower bound in lexicographically ordered C.

**Physics Forums | Science Articles, Homework Help, Discussion**