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Riemann Sphere Absolute Values

TOPO-7BFMCC

Which of the following nets converge(s)? Select ALL that apply.

A

The absolute value function $|\cdot|:\mathbb C\rightarrow\mathbb R_{\ge 0}$ sending $a+bi\mapsto\sqrt{a^2+b^2}$ where $\mathbb C$ as a directed set has the partial order given by comparing absolute values.

B

The absolute value function $|\cdot|:\mathbb C\rightarrow\mathbb R_{\ge 0}$ sending $a+bi\mapsto\sqrt{a^2+b^2}$ where $\mathbb C$ as a directed set has the partial order reverse to that given by comparing absolute values.

C

The absolute value function $|\cdot|:\mathbb C-\{0\}\rightarrow\mathbb R_{>0}$ sending $a+bi\mapsto\sqrt{a^2+b^2}$ where $\mathbb C-\{0\}$ as a directed set has the partial order reverse to that given by comparing absolute values.

D

The absolute value function $|\cdot|:\mathbb C\cup\{\infty\}\rightarrow[0,\infty]$ sending $a+bi\mapsto\sqrt{a^2+b^2}$ and $\infty\mapsto\infty$ where $\mathbb C\cup\{\infty\}$ as a directed set has the partial order given by comparing absolute values.