Is it true that $\{a\}$ is closed but not open for any real number $a$?
For me, I could first let a set $A=\{a\}$,
and let $C=\{\text{collection of all subsets of }A\}$.
Since the definition of topological space requires that both the empty set and $A$ belong to $C$, thus I can conclude that $A$ must be open.
Now, $A$ is closed since its complement, the empty set is open.
Therefore I see it's both open and closed.
Best Answer
"Open" and "closed" are not absolute terms, they are relative terms. A subset of a set is "open" with respect to a particular topology, and "closed" with respect to a particular topology.
The real numbers have lots of topologies you can place on them. However, there is one topology which is used most often, namely, the "standard topology". This is the topology that arises from the metric, which makes a subset $\mathscr{O}\subseteq\mathbb{R}$ open if and only if for every $x\in \mathscr{O}$ there exists $\epsilon\gt 0$ such that $$B(x,\epsilon) = \{r\in\mathbb{R}\mid |r-x|\lt\epsilon\} \subseteq \mathscr{O}.$$
When one asks if a subset of $\mathbb{R}$ is "open" or "closed" and does not specify a topology for $\mathbb{R}$, it is assumed that one is talking about the standard topology of $\mathbb{R}$, the one described above.
So your question is not asking "Can you come up with a topology on $\{a\}$ that makes it open or closed?" (The answer to that question is always "yes", no matter what the set you are looking at is). The question you are asking is really:
So, use the definition of "open" for the standard topology to see whether $\mathbb{R}-\{a\}$ is open (that will tell you whether the set is closed in the standard topology); and use the definition of "open" for the standard topology to see whether $\{a\}$ is open.