Q:

Are finite sets in R open?

I'm self-studying general topology and getting incredibly confused with the notions of open and closed sets. 
In the following, I'm considering the space X = $\Re$, the set A = {2}, the collection $\tau$ of subsets of $\Re$: $\tau$ = {∅, $\Re$, {2}}. Since $\tau$ satisfies the axioms of a topology, it is a topology. And since A belongs to $\tau$, it is an open set. 
However I'm also aware of the proof that finite sets are closed in $\Re$, using the argument that the complement of the singleton set is open. Where did I go wrong with my argument?

A:

Note that the question in your title is incomplete. This is because we can't discuss "open sets of $\mathbb{R}$" without mentioning the topology --- A more pedantic (which may be useful at this stage of your learning) way of asking this is "Are finite sets in $(\mathbb{R},\{\emptyset,\mathbb{R},\{2\}\}$ open?". This is because to talk about open sets, we need a topological space, which is both the set ($\mathbb{R}$ here) and the topology on that set ($\{\emptyset, \mathbb{R},\{2\}\}$ here). 
When the topology chosen is "standard", this second part can be omitted --- A question such as "What are the compact sets in $\mathbb{R}$?" would implicitly assume that the "standard topology" on $\mathbb{R}$ is being used, meaning the metric topology given by the euclidean norm. If you aren't considering this topology, it's imperative to mention it. I know you do in the body of the question, but as it appears you're new to this field, I just wanted to reinforce that both parts are equally important.
Now, for your actual question:
Open sets are by definition any element of $\tau$. Consider the finite set $\{1\}$ --- is this in $\tau$? What about $\{2\}$? If every finite set is in $\tau$, then every finite set is open. In this way, $\tau$ tells you precisely what sets are open tautologically. Especially in a case like this (where $\tau$ is written down as an enumeration of its elements) it should be rather easy to check. This will change later when you instead talk about topologies generated by some set of elements (So a certain subset of the topology is written down, and then you essentially say "add in everything else to make the topology satisfy the closure properties required to be a topology"). The relevant terms here are basis and sub-basis, but I expect you'll learn these in the next few weeks.

