My Topology book says that in $\Bbb{R^2}$, the set of all one-dimensional subspaces (or lines passing through the origin) is a circle.
This stackexchange question says that this is because every subspace can be uniquely identified by the points at which it intersects the circle. I have the following questions:
If uniquely identifying each subspace by its intersection points is our purpose, then wouldn't a semicircle on the x-axis do as well?
It seems a set containing unique (symbolic) representations for each subspace is the same as the set of subspaces. Is this interpretation correct? In countable cases, would a set of the form $\{a_1,a_2,\dots\}$ be the same as the set of subspaces?
Your first question is basically answered by @DanValenzuela's comment; you can represent this collection of subspaces as a semicircle, provided you identify the two endpoints of the semicircle (since they intersect the same linear subspace). However, until you place a topology on the collection $X$ of linear subspaces of $\mathbb{R}^2$, we cannot quite say $X$ is homeomorphic to $\mathbb{RP}^1$; we've only determined a bijection of sets.
As for your second question, one does not literally mean that the two sets are the same. When working with one space (such as the collection of linear subspaces of $\mathbb{R}^2$), we often think in terms of a different, more convenient set (such as their intersection points with $S^1$, or $\mathbb{RP}^1$). For example, we refer to angles as "$\pi/3$" or "$7\pi/2$" or "$513^\circ$". Even though we are picturing geometric angles, we refer to them as elements of $\mathbb{R}$ for convenience.
In the above cases, we are taking advantage of set bijections. But there are other -- sometimes stronger -- forms of equivalence that are useful in other areas of math. For example, topologists will commonly call an object a "sphere" or a "ball", when they really mean that the space is homeomorphic to a sphere or a ball. In the case of $X$ above, the most natural topology to put on $X$ is the one that makes it homeomorphic to $\mathbb{RP}^1$. So a topologist would say that $X$ and $\mathbb{RP}^1$ are the same. The language all depends on the category of mathematical objects you're using and the notion of equivalence in that category.