Shrinking open interval empty? Let (a, b) an open interval in $\Bbb R$.
If $\lim_{b\to a}$, then is the resulting interval empty?
So my intuition is that it is empty, but since b 'approaches' a, I'm not sure if ever b reaches a. 
Here's why I think it's empty:
$$(a, b):=\{x | a<x<b\},$$
$$\lim_{b\to a}b=a,$$
so clearly,
$$\lim_{b\to a}(a, b)=\{x | a<x<a\}=\emptyset.$$
 A: Yes - although no particular interval "along the way" is empty, the "limit" of the intervals is the emptyset.
Another good example of this is to let $X_a=(a, \infty)$. Then each $X_a$ is infinite, but the limit of the $X_a$s is empty! Do you see why?

We should be careful here: it's worth giving a precise definition of "the limit of a set." The one that makes the most sense is $$x\in\lim_{a\rightarrow b}X_a\quad \iff\quad \exists\delta>0\forall a[0<\vert a-b\vert<\delta\implies x\in X_a],$$ that is, $x$ is in the limit of the $X_a$s (here $X_a=(a, b)$) iff $x$ "eventually stays in" the $X_a$s. (In your example you probably want to tweak this a bit: since you're looking at $a<b$, you only want the "left hand" limit.)
However, you need to be extremely careful with this definition! For example, it is not the case that (using this definition) the complement of the limit is the limit of the complements! For example, let $X_a=\emptyset$ if $a$ is rational, and $\mathbb{R}$ if $a$ is irrational. Then 


*

*What is $\lim_{a\rightarrow 0}X_a$?

*What is $\lim_{a\rightarrow 0}X_a^c$? (Here "$^c$" denotes the complement.)
What's going on here is that the sets are not changing monotonically: as $a\rightarrow 0$ points are entering $X_a$, then leaving $X_a$, then entering again, etc. In the example you give, everything is nicely monotonic: if $a\le a'<b$, then $(a, b)\supseteq (a', b)$. But in general, the limit of sets is not something which behaves intuitively.

Two more comments:


*

*Note that in your question - and in my answer - we've talked about sets indexed by reals. We can also talk about the limit of a sequence of sets, that is, sets indexed by natural numbers; or other ways of indexing sets. 

*Thinking about "limits of sets" might make you think: hey, wait a second! Limits in calculus make sense because of the geometry (or topology) of the real line; if I want to talk about limits of sets, do I need to have a geometric picture of sets of real numbers? The answer is: yes, you probably should, and you can! See e.g. https://en.wikipedia.org/wiki/Hypertopology.
A: Yes, but the way I would put the question is, whether the set
$$
\cap_{b > a} (a, b)
$$
is empty.  And it is: whatever the number $c > a$, there exists a $b$ such that $a < b < c$, so that $c$ is not in $(a, b)$ for that particular $b$, hence is not in the above intersection.
Finally, $a$ is not in the above intersection because it is not contained in any of the intervals $(a, b)$.
