I'm looking at exercise 4 in page 12 of Gamelin's Introduction to Topology.
The problem is stated as follows:
Suppose that $F$ is a subset of the first category in a metric space $X$ and $E$ is a subset of $F$. Show by an example that $E$ may not be of the first category in the metric space $F$.
The solution to the exercise says that $\Bbb R$ is of the first category in $\Bbb R^2$ but not of the first category in itself.
I understand the part that $\Bbb R$ is not of the first category in itself. I think it is since every singleton subsets of $\Bbb R$ is nowhere dense but the countable union of the sets cannot be $\Bbb R$ since then $\Bbb R$ must have empty interior, which is clearly false.
However, I don't see how to prove that $\Bbb R$ is of the first category in $\Bbb R^2$. How can I show this?
That $\mathbb{R}$ is not of the first category doesn't follow from the fact that singletons are nowhere dense but because it is a complete metric space and the result follows from Baire's category theorem. However, when seeing $\mathbb{R}$ as a subset of $\mathbb{R}^2$, note that a line is closed and has empty interior so it is itself nowhere dense. Writing $\mathbb{R}$ as an union of itself makes it of the first category in $\mathbb{R}^2$.
Baird's category theorem (one form of it) states that every complete metric space is a Baire space. Although the definition of Baires space states that for every countable collection of dense open subsets $(U_n)$ their intersection $\bigcap U_n$ is dense, an equivalence is that every open subset is of the second category. With this in mind, note that $\mathbb{R}$ is an open subset of itself, so it is of second category.
For your second comment, it is true that $\mathbb{R}$ is not properly a subset of $\mathbb{R}$, but we are thinking of it as homeomorphic to a subset of $\mathbb{R}^2$, say, the set $\{(x,0)\in\mathbb{R}^2\mid x\in\mathbb{R}\}$. You can see this subset as a metric space with the metric that it inherits from $\mathbb{R}^2$ and then you obtain a space that is quite similar to $\mathbb{R}$. Is something similar as saying that $\mathbb{R}$ is contained in the complex numbers $\mathbb{C}$.