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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:

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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.

Alooffi
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  • Welcome to MSE! <> 1. The topologies are the Euclidean topologies. 2. If you'll grant the square is homeomorphic to the closed unit disk, then embedding the disk as the closed upper unit hemisphere gives the desired homeomorphism without much fuss. This is surely answered on site, so it's worth searching if you haven't already. :) – Andrew D. Hwang Aug 01 '21 at 20:21
  • Thanks Andrew! The link does include an explicit homeomorphism between the disk and the hemisphere as you described, so I'm covered on that front xD. That being said, I still wonder if there's a shorter "path" than jumping though all those spaces, or if at least there's some intuition as to how one would decide to try going from $\mathbb{RP}^2$ to the upper hemisphere, then to the disk, and then to the square. – Alooffi Aug 02 '21 at 03:43
  • Also, a bit of a pedagogical question: Would it be useful for someone at my stage to explicitly construct the bijections and establish their continuity, or would it be better try to develop some intuition regarding the maps? For example, for square to disk I thought of using lines through the center and let a function take the intersection line/square to the intersection line/circle. "Clearly" it is a bijection and takes open sets to open sets in both directions and so it would be a homeomorphism, but would I benefit from actually writing the function down? – Alooffi Aug 02 '21 at 03:46
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    It's always hard to say "there's no shorter path", but offhand none comes to mind. :) The hemisphere picture is a "slice" of the group action on $\mathbf{R}^3\setminus{0}$, so geometrically natural. <> In my experience, geometric and topological intuition versus analytic arguments and algebraic formulas are complementary. It's useful to have one when the other is lacking. Here, writing an explicit homeomorphism from the square to the disk does not strike me (personally) as that important. Understanding the boundary gluing, however, does. – Andrew D. Hwang Aug 02 '21 at 11:07
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    autodavid's answer here is probably of interest. – Andrew D. Hwang Aug 02 '21 at 11:09
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    Thanks once again, very helpful advice! About the answer you linked... whoever said making math diagrams is an art was absolutely correct. – Alooffi Aug 03 '21 at 12:19
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    You might enjoy A Topological Picturebook by George K. Francis. :) – Andrew D. Hwang Aug 03 '21 at 14:02

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