Why can't I apply subspace topology from $\mathbb{R}^3$ on real projective plane

Question

In Crossley's Essential topology, the line about $\mathbb{R}P^2$ - It is not a subset of $\mathbb{R}^3$, since the elements of $\mathbb{R}P^2$ are not points in $\mathbb{R}^3$ but subsets of $\mathbb{R}^3$

I am not getting the point once it is said $\mathbb{R}P^2$ is not a subset of $\mathbb{R}^3$ but elements of it are subsets of $\mathbb{R}^3$, what does it mean?

Basically, I am asking why can't I apply the subspace topology of $\mathbb{R}^3$? Can anyone please explain?

Answer 1

$\Bbb RP^2$ is defined as a quotient space. The canonical construction of a quotient space is to let its elements be equivalence classes. In this case, these equivalence classes are actually sets of elements of $\Bbb R^3$, specifically, the point $[x]\in\Bbb RP^2$ is the set of all $y\in\Bbb R^3$ with $y=\lambda x$ for some nonzero $\lambda\in\Bbb R$.

You can't "apply the subspace topology" because there is no natural way to view $\Bbb RP^2$ as a subset of $\Bbb R^3$ and its topology is the quotient topology, not the subspace topology. Perhaps there is a fancy embedding $\Bbb RP^2\hookrightarrow\Bbb R^3$, perhaps not - I wouldn't know. But thinking of it as a subspace (from a subset) is not right.

Answer 2

Each element of the set $X = \{\{A,B\},\{B,C\},\{A,C\}\}$ is a subset of the set $\{A,B,C\}$, but $X$ itself is not a subset of $\{A,B,C\}$.

Similarly, each element of the set $\mathbb RP^2$ is a subset of the set $\mathbb R^3$, but $\mathbb RP^2$ itself is not a subset of $\mathbb R^3$. And to apply the subspace topology, you are required to start with a subset of $\mathbb R^3$.