A bit of context: I'm starting to learn topology using Topology: A Categorical Approach and the quotient topology is defined as being the finest topology on a set $S$ for which the surjective map $\pi : X \rightarrow S$ is continuous; as well as in terms of a universal property:
In the examples, the authors define $\mathbb{RP}^2$ as the quotient of $\mathbb{R}^3 \backslash \{0\}$ by the equivalence relation $x \sim \lambda x$ for $\lambda \in \mathbb{R}$. They also give a second definition of it, as the quotient of the unit square $I^2$ by the equivalence relation $(x,0) \sim (1-x,1)$ and $(0,y) \sim (1,1-y)$. It's not an exercise, but they do encourage the reader to justify these two definitions by showing the spaces are homeomorphic.
My first doubt is: the quotient topology depends on the topology given to the original set... so what are the topologies on $\mathbb{R}^3 \backslash \{0\}$ and $I^2$? Are both using the metric topology? I'm guessing that one gets used to knowing which topology is the "standard" for a given space after studying the topic for some time, but I'm still not quite there yet.
About the actual problem: is there a way to show these two are homeomorphic without having to "jump" to a third space? After researching for a bit, the method I understood the best was to first show the second definition is homeomorphic to the quotient of the unit disk $\mathbb{D}^2$ by the equivalence relation $(x,y) \sim (-x,-y)$ for $(x,y)$ in the boundary. See: https://courses.maths.ox.ac.uk/node/download_material/14654 (which incidentally makes no mention of the topologies at play either). However, putting it all together would make the proof quite long, since the argument for $\mathbb{RP}^2 \simeq \mathbb{D}^2/\sim $ given in the link depends on yet another homeomorphic space.
Given that by this point in the book I can't say I know much topology nor do I have a particularly extensive collection of examples to develop some intuition yet, I'd like to know if there's something much more "basic" that I'm missing.