Does $[a,\infty]$ have a meaning? We know that $[a,b]$ denotes a closed interval. And $(a,b)$ the open one. And we often denote the positive real numbers by, $\{x:x\in(0,\infty)\}$. But I remember I somewhere saw this: $$[0,\infty]$$
But what I cannot understand is, that how can infinity be included in an interval. Because infinity is not a number, not defined. So how can it be included in an interval? If it is included, then what is the difference between $\infty]$ and $\infty)?$
 A: These two elements $-\infty, +\infty$ are called extended reals.  It would be better, in your examples, to write the symbol with a sign:  $+\infty$. These are just elements that are adjoined to the reals to provide an endpoints before and after all the finite reals. They're a matter of convenience only. 
The interval $[-\infty, +\infty]$ is homeomorphic to any other compact interval $[a,b]$ (for $a < b$). The difference between $[0,+\infty)$ and $[0,+\infty]$ is just like the difference between $[0,1)$ and $[0,1]$.
$±\infty$ is/are not to be confused with "NaN", alias "not a number".
Sometimes a single element called by convention $\infty$ (no sign) is adjoined a topological space such as $\mathbb{R}$ in order to form the one-point compactification, but that's a different notion and is not generally what is meant in discussions of calculus and analysis.
A: The extended real line, often written as $\overline{\Bbb R}$, is defined to be $\Bbb R\cup\{-\infty,+\infty\}$.
Here, $-\infty$ and $+\infty$ can be thought of as meaningless symbols. The extended real line is just the regular real line, plus these two symbols.
Now, usually, the relation $\le$ is extended a bit when dealing with $\overline{\Bbb R}$. We define $a\le+\infty$ to be true for all $a\in\overline{\Bbb R}$, and we define $-\infty\le a$ to be true for all $a\in\overline{\Bbb R}$. These are just new definitions we made for convenience. Similarly, the operators $+$, $-$, $\times$, and $\div$ are usually extended when dealing with this set.
The set $[a,+\infty]$ is just the subset of $\overline{\Bbb R}$ consisting of everything bigger than or equal to $a$, including $+\infty$. (Note that we have $\overline{\Bbb R}=[-\infty,+\infty]$.)

Here's an example of when such a set could be useful. There's a theorem (called the Extreme Value Theorem, or EVT) that says that, if $f$ is a continuous function with domain $[a,b]$, then $f$ is bounded on $[a,b]$.
Now, let's say we have a continuous function $f$ with domain and codomain $\Bbb R$. In addition, let's say:
$$\lim_{x\to-\infty}f(x)=\lim_{x\to+\infty}f(x)=0$$
Can we show that $f$ is bounded?
Sure! Let's extend the domain of $f$ to $\overline{\Bbb R}$, by defining $f(-\infty)=f(+\infty)=0$. This new, extended version of $f$ is still continuous, because of the limits given above. So now we have a continuous function $f$ with domain $\overline{\Bbb R}=[-\infty,+\infty]$. By the EVT, $f$ is bounded. QED.
