Confusion. about the precise definition on $\infty$ in complex analysis

Question

In the complex analysis, we define a point $\infty$ on the complex plane. Then the textbook claims that (a) None of the semi-plane (all the points on one side of a fixed line) contains the $\infty$ (b) every line goes through the $\infty$. But (b) implies that $\infty \in l$, $\forall$ line $l$. And we can now pick a line contained in the semi-plane by letting the line be parallel to the boundary of the semi-plane. Then $\infty$ is a member of the semi-plane, contradicts what (a) said. But what is wrong in the above argument? And what's the precise definition of $\infty \in S$, $S$ a subset of $C\cup \{\infty\}$?

Answer 1

(a) and (b) are really definitions, though they may not be phrased that way. The textbook is saying that when they use the word "line" in discussing a subset of $\mathbb{C} \cup \{\infty\}$, it will include $\infty$ (i.e., it will include an ordinary line in $\mathbb{C}$ together with $\infty$), while when they use the word "semi-plane" it will not include $\infty$ (it will include just the ordinary semi-plane in $\mathbb{C}$).

If you want a precise description of how $\infty$ is included in the complex plane, look up "stereographic projection" of the sphere to the plane. But that doesn't really change the fact that (a) and (b) are just definitions of how we use the words "line" and "semi-plane" in the context of $\mathbb{C} \cup \{ \infty \}$.

For a justification for why this is a reasonable definition, you can think that since a line includes $\infty$, then a semi-plane is one component of the complement of the line, so the semi-plane will not include $\infty$. This does have the consequence that a line (extended to $\mathbb{C} \cup \{ \infty \}$) parallel to the boundary of a semi-plane, will no longer be contained in the semi-plane (which is really what your argument shows), which may seem weird, but there is no logical contradiction.

Answer 2

The Riemann sphere is the one-point campactification of the complex plane (or $\Bbb R^2$).

It's well known that the standard $2$-sphere $S^2$ minus a point is homeomorphic, indeed diffeomorphic, to $\Bbb R^2$.

Anyway $\infty $ can be thought of as the added (or subtracted) point.