What is the definition of $\infty$ I saw in a note that say $\infty$ is not a real number, and there is no interval of the form $(a, \infty]$?
So what is the definintion of $\infty$?
 A: The symbol "$\infty$" is part of the ordinary mathematical alphabet. Like many other symbols in the mathematical alphabet, it has various uses. There is no standalone definition of $\infty$.  There are, however, precise definitions of various expressions that use the letter $\infty$. 
In particular, in analysis, there is a definition of what is meant by the interval $(a,\infty)$. Note that there is nothing peculiar about the fact that there is no standalone definition of $\infty$, while some expressions that use $\infty$ have a clear conventional meaning. There is no standalone definition of $a$ either, or of $($. 
Comment: Perhaps the following analogy will be useful. Suppose that we have written a program that finds, for any $n$, the $n$-th decimal digit of $\pi$. Then the command "Print $[17, 42]$" might mean print all the digits from the $17$-th to the $42$-th, inclusive. The command "Print $[17, 42)$" could be used to mean print from the $17$-th (inclusive) to the $42$-th (exclusive).  And if we wanted to print all the digits from the $17$-th on, we might issue the command "Print $[17,\infty)$".  But the command "Print $[17,\infty]$ would make no sense, no reasonable meaning can be assigned to the phrase "the $\infty$-th digit of $\pi$."
A: You could also use $\infty$ as the point that compactifies $\mathbb{C}.$
Thus, it is convenient to use the point at infinity when constructing the Riemann sphere. So, in topology, it is used to  compactify certain topological spaces.
A: There are a variety of different concepts of infinity in mathematics.  When used in interval notation of this kind, something like $(5,\infty)$ means the set of all real numbers greater than $5$.  Similarly $[5,\infty)$ means the set of all real numbers greater than or equal to $5$.
