What is the realcompactification of the real line? I've studied the definition of the Stone-Čech compactification by Munkres Topology. I have realized that we can't write the Stone-Čech compactification of the real line explicitly, we are just able to build it and show that it is compact. Am I right?
What is  the realcompactification exactly? Wikipedia says:
"In mathematics, in the field of topology, a topological space is said to be realcompact if it is completely regular Hausdorff and every point of its Stone–Čech compactification is real (meaning that the quotient field at that point of the ring of real functions is the reals). "
Four questions arise naturally:

*

*What does the statement above mean?

*Can we obtain the realcomapctification of the real line explicitly?

*Are there any criteria for understanding if a point belongs to its realcompactification, but not to its Stone-Čech compactification?

*Is the realcompactification compact too?

I would be so grateful if you name a book which explains this sort of compactification perfectly.
Thank you in advance.

EDIT: I have found another definition,
The realcompactification of $X$ is the largest subspace of its Stone-Čech compactification in which $X$ is $C$-embedded.
According to this, how can I conclude that the real line is realcompact?
On the other hand, by Wikipedia:
"The (Hewitt) realcompactification υX of a topological space X consists of the real points of its Stone–Čech compactification βX. A topological space X is realcompact if and only if it coincides with its Hewitt realcompactification."
According to this, it's obvious that the real line is realcompact.
 A: Realcompactness is a not very well-known property of topological spaces; it does not come up in elementary topology courses, usually. 
I'll quote some of the results in paragraph 3.11 of Engelking's (IMHO very good) standard work General Topology (revised and completed edition). Gilman and Jerrison also cover it in Rings of Continuous Functions.
The "internal" way of defining it is not very hard, but somewhat technical and sort of explains its name (at least the compactness part): $X$ is realcompact iff it is Tychonoff (i.e. completely regular Hausdorff) and every ultrafilter in the poset of functionally closed subsets of $X$ that has the countable intersection property has non-empty intersection.
(A subset of $X$ is functionally closed iff it is of the form $f^{-1}[\{0\}]$ for some continuous $f:X \to \Bbb R$, these sets form a poset ordered by inclusion and a filter in it is a family of functionally closed subsets that is closed under finite intersections and supersets and does not contain $\emptyset$; an ultrafilter is a maximal (by inclusion) filter; all as usual. The Cech-Stone compactification has as points essentially all ultrafilters in the functionally closed sets).
From this it follows easily that any regular Lindelöf space is realcompact, so $\Bbb R$ is too. 
An external way of viewing the property: $X$ is realcompact iff $X$ is homeomorphic to a closed subspace of $\Bbb R^I$ (for some index set $I$, in the product topology). This makes it obvious that realcompactness is preserved by all products (like compactness) and is inherited by closed subspaces (also like many compactness properties). Also, trivially (this way) all $\Bbb R^I$ spaces, including the reals itself, are realcompact.
The formulation as under 1 is more in the vein as Gilman and Jerrison would put it (in terms of the ring of continuous functions on $X$). I could expand on that if you have some background in the theory, otherwise it would lead us somewhat astray.
The (Hewitt) realcompactification $\nu X$ of $X$ is the maximal subset of $\beta X$ such that every continuous $f:X \to \Bbb R$ has an extension (unique of course) to $\nu X$. If $X$ is already realcompact this equals $X$ of course. $\beta \Bbb R$ is indeed hard to describe, as you said, but $\nu \Bbb R=\Bbb R$ is easy.
The property is really quite natural if you're looking at the ring of (real-valued) continuous functions $C(X)$, just as the Cech-Stone compactification comes up quite naturally: $\beta X$ is the compact space such that $C^\ast(X)$ (the bounded continuous real-valued functions) and $C(\beta X)$ are ring-isomorphic and $\nu X$ has the property that $C(\nu X)$ and $C(X)$ are ring-isomorphic. 
The fact "$X$ is compact iff it is countably compact and Lindelöf" has a nice parallel in "$X$ is compact Hausdorff iff it is realcompact and pseudocompact" (as the Wikipedia page also mentions).
