Does $J$ represent finite set in most occasions? There are some commonly used letters that have the same meaning in topology, like:

*

*$I$ represent index set

*$G,U$ open set

*$F$ closed set

*$K$ compact set

*$N$ neighborhood

But are there any commonly used letters for finite, countable and uncountable sets?
 A: Finite sets are often represented by $S$, countable sets by $N$ and uncountable sets by $A$, $I$ or $\Gamma$ depending on who is doing the writing.
Much of the standard set notation that you describe comes from the mathematicians working in early Set Theory -- Cantor, Hausdorff, Bolzano -- and topology (or Analysis Situs as it was first known) -- Poincaré and so the traditional letters tend to stand for French or German terms.  In particular:

*

*G -- Geöffnete ('open' in German)

*F -- Fermé     ('closed' in French)

*K -- Kompakte  ('compact' in German)

*U -- Umgebung  ('neighbourhood' in German)

Using $N$ for neighbourhood appears to be a later innovation and might (I am not an expert on this history of this) have become more common after ${\mathbb N}$ started being used for the natural numbers since there is a potential for confusion otherwise.
$A$ is a capital letter $\alpha$ rather than a Latin $a$ and appears to be used as lower-case Greek letters are used for ordinals ($\alpha$, $\beta$) which in turn are typically used to index nets in topological spaces (which generalise sequences).  $S$ simply stands for set as you will have guessed and seems to be used for finite sets as introductions to set theory (e.g. Jech) usually start with finite sets called $S$.
$I$ and $J$ are both used to indicate uncountable ordered index sets (typically there is little wrong in considering $I$ to stand for interval as well) and $J$ is used because it is the successor of $I$ whenever we need to sub-interval it.
$\Gamma$ gets used as the uncountable index set by functional analysts (e.g. $c_0(\Gamma), l_1(\Gamma)$ and $l_\infty (\Gamma)$) though I've not encountered any reason for this choice so far (Day's book uses it and undoubtedly helped popularise it, but doesn't explain it).
As always with notation though, provided you clearly explain what you're intending by it (at the start) and you're not obviously abusing it ("Let $n(\varepsilon)$ be small and approach $0$ as $\varepsilon$ grows without bound..."), you are free to choose your own, and maybe even create a standard.
