$\mathbb{C} \cup \{ \infty \}$ and $\mathbb{R} \cup \{ -\infty, +\infty \}$ On the first page of the textbook Mathematical Analysis For Machine Learning and Data Mining by Dan Simovici, the following table of notation is presented:

$\mathbb{R}_{<>0}$ seems to be a typo, but that isn't my question.
My question relates to $\mathbb{\hat{C}}$ being described as "the set $\mathbb{C} \cup \{ \infty \}$", whereas $\mathbb{\hat{R}}$ is described as "the set $\mathbb{R} \cup \{ -\infty, +\infty \}$". I'm curious as to why, for the complex number case, the union involves the set containing only positive infinity, whereas the union for the $\mathbb{\hat{R}}$ case involves the set containing both positive and negative infinity? Do the complex numbers have some property that makes having $-\infty$ redundant when it has $\infty$? Why this difference between the complex number case and the real number case?
I would greatly appreciate it if people would please take the time to clarify this.
 A: "Complex infinity" is near to numbers of large magnitude in $\mathbb{C}$. The extended complex numbers $\mathbb{C}\cup\{\infty\}$ can be visualised as a sphere (called the Riemann Sphere) with the two poles being $0$ and $\infty$. Say $\infty$ is the north pole. Then complex numbers of large magnitude are near to the north pole, including large negative reals.

A: In the real line you have two directions from any point: the positive and the negative direction. In the complex plane there are infinitely many directions. The point at $\infty$ in the complex plane  is not along any particular direction (and you cannot call it it 'positive infinity'). 
A: Firstly, this is just what he has considered useful and will use in his book.  Some of those notations are very common and used by many but some are less common.  E.g. I might not have guessed the meaning of $\mathbb{I}$ without this table. 
Both $\mathbb{R}$ and $\mathbb{C}$ can be extended with a single point typically denoted by $\infty$.  In the first case, it is topologically equivalent to a circle and the second a sphere.  It is probably best not to think of this as "positive infinity" but just one additional point.  
In the case of $\mathbb{R}$, you have another obvious choice of the two point extension with $+ \infty$ and $- \infty$.  Now, it is topologically equivalent to a closed interval e.g. $[-1, +1]$.  However, with $\mathbb{C}$, there are far more than $2$ directions towards infinity, how about $i \infty$ and $-i \infty$ or more generally $(\cos \theta + i \sin \theta) \infty$?  This cold be done but it seems that it is more rarely found useful or interesting.  
(When I say "topologically equivalent", I mean with the most commonly used topology.  Other topologies are possible.)
A: Adding infinities amounts to defining certain sets of "neighbourhoods" of those infinities, which then let you speak about convergence, limits, topology, etc.
In real analysis, you define neighbourhoods of $-\infty$ as the sets containing one of the intervals $(-\infty, a)$ for $a\in\mathbb R$, and similarly you define neighbourhoods of $+\infty$ as the sets containing $(a, +\infty)$ for $a\in\mathbb R$.
In complex analysis it is convenient to define one single infinity $\infty$ by defining its "neigbourhoods" as sets containing $\{z:|z|>a\}$ for $a\in\mathbb R$, usually $a\ge 0$ (i.e. sets containing exteriors of circles centred in $0$). This is not to say it won't be possible to define other sorts of neighbourhoods (and therefore add infinities in a different way), but the construction with one single infinity added is, by large, the most basic and useful of all. For example, it lets you conclude that $\lim_{z\to 0}\frac{1}{z}=\infty$ and $\lim_{z\to\infty}\frac{1}{z}=0$.
A: There are two commonly used "compactifications" of number sets, the "one point compactification" and the "Stone-Cech compactification".
The "one point compactification" of the complex numbers is the sphere $C\cup\infty$.  The "Stone-Cech compactification" adds a different "infinity" at each "end" of a line through 0 and gives the topology of a disk.  The "one point compactification" is more often used since there is only one "point at infinity" rather than an infinite set of them.
The "one point compactification" of the real numbers (or any open interval) adds a single "point at infinity" that is "arbitrarily close" to points near both ends.  It gives the set the topology of a "ring" or "closed loop".  The "Stone-Cech compactification" adds two "points at infinity", $\infty$ and $-\infty$, and gives the set the topology of a closed interval.  It is more commonly used for the real numbers because the topology of all the "finite" real numbers stays what we are used to- a line rather than a loop.
