Understanding a definition of limit of a set $X$ in real analysis I was studying about limits in real analysis (from a book Problems of Calculus in One Variable written by I.A Maron) when , I came accross the definition :

A point $a$ on the real axis is called the limit point of a set $X$ if any neighborhood of the point $a$ contains points belonging to $X$ which are different from $a$ ($a$ may either a proper or an improper point) .

However, I don't get what is meant by $real\space axis$ ,I am considering it as $X$ axis (by drawing an analogy to limit of functions , as it was written on that particular section in the book .) Now, I think this assertion needs a slight modification upon the part:" contains points belonging to $X$ which are different from $a$". This may not be the case if we consider the sequence :$2,2,2...$, right? Thus the number $a$ might be in the neighborhood .Also, I dont understand what is a proper or an improper point ?
 A: The definition (not "assertion") of your book for "limit point of a set of real numbers" is a particular case of "limit point of a subset in a topological space", and needs no "slight modification".
I think your quote is not completely faithful: it should begin with "A point [...] is called a limit point of a set", not the limit point. It is rarely unique. For instance, the set of limit points (also called the "derived set") of $X=(1,2)\cup\{3\}$ is $[1,2].$
This notion must not be confused with those of limit of a sequence or a function. The set $X$ is not a sequence.
The "real axis" is the set $\Bbb R$ of real numbers.
What your book calls "improper point" is probably "real number not belonging to $X$". In that sense, the two limit points $1$ and $2$ of the previous example are "improper".
Another (though less probable) interpretation could be that the author wants to include $+\infty$ and $-\infty$ as "improper" real numbers. In the topological space $[-\infty,+\infty],$ the set of limit points of $(0,+\infty)$ is $[0,+\infty].$
