Exercise 18.3 from James Munkres' Topology, how should I interpret this question? 
3. Let $X$ and $X'$ denote a single set in the two topologies $\mathcal T$ and $\mathcal T'$, respectively. Let $i:X'\to X$ tbe the identity function.
  (a) Show that $i$ is continuous $\Leftrightarrow \mathcal T'$ is finer than $\mathcal T$.
  (b) Show that $i$ is a homeomorphism $\Leftrightarrow \mathcal{T'=T}$.

It's something about the first sentence that confuses me.
Okay, what I thought was: $X\in\mathcal T$ and $X'\in\mathcal T'$, 
where $X$ and $X'$ are (open) single sets, which I interpret as open sets having one element.
But I guess my interpretation is wrong, as the question doesn't make much sense in this way.
So my question is how to interpret this question, and why I should interpret it this way.
Don't spoil the answer!
 A: He means that $X$ and $X'$ are two topological spaces with the same underlying set. $X$ carries the topology $\mathcal T$ and $X'$ the topology $\mathcal T'$.
I would have formulated it in the following way:
Let $\mathcal T$ and $\mathcal T'$ be topologies on a set $X$ and consider the function
\begin{align*}
i\colon (X,\mathcal T')&\to (X,\mathcal T)\\
x&\mapsto x.
\end{align*}
That way I would also have avoided to call $i$ the identity function; a true identity should always be continuos.
A: What's meant is that the toplogies $\mathcal{T}$ and $\mathcal{T}'$ have a set, called both $X$ and $X'$ for some seemingly unnecessary reason, in common.
Regarding why it should be interpreted this way; well I guess that's just what it says in English.
A: I think, what is meant that you have two topological spaces $X = (X, \mathcal{T})$ and $X' = (X', \mathcal{T}')$ where $X=X'$ as sets, but not necessarily as topological spaces.
As for why: The question (b) means that it should be possible to make a statement about the equality of $\mathcal{T}$ and $\mathcal{T}'$ using "information" from $i$. Therefore $i$ should "use" the whole space, not only a subset. In other words if $i$ operated on a small subspace of the topolgical space, you would hardly be able to make a statement on the whole topology.
