Connectedness
Twilight Sparkle and Sunset Shimmer found themselves curled up beside each other, sitting on the edge of Sunset’s bed. As Twilight played with her glasses, infodumping on about point set topology, Sunset merely enjoyed the ambience, radiating in the warmth of the overexcited girl beside her.
“Oh!” Twilight halted her rambling momentarily. “That reminds me of something I meant to give you.” She brushed a lock of hair out of her field of vision and curled it around her ear. “After all, I really like topology, and I really like you.”
“Thanks, I like you too, dork.” Sunset pointed finger guns at Twilight. “Though I am still surprised you of all people like topology so much.”
Twilight scrunched up her face. “What do you mean?”
Sunset bit the inside of her cheek, forming a little dimple on the outside. “It’s the TERFiest branch of mathematics, isn’t it? Reducing things to just their number of holes?”
Twilight shook her head, raised her index finger, and scoffed, “First of all, you did not just insinuate an entire branch of mathematics can be transphobic.”
Sunset placed her hand under chin, lowering her head slightly as she whispered rather loudly, “I think I did.”
Twilight laughed under her breath and raised another finger. “Second, human male and female bodies are topologically equivalent. Same number of holes.”
Sunset poked her thumb under the rim of her underwear over her shirt, raising an eyebrow.
Twilight shook her head. “Topologically speaking. The anatomical difference does not meaningful affect the geometry of the object. Er, person. Note that sex reassignment surgery can be viewed as a continuous deformation, stretching and folding without cutting and pasting.” Twilight grimaced. “Though that’s a weak argument, when said deformation does require a scalpel physically if not geometrically.”
Sunset winced. “I did not need that visual in my head.”
“Sorry.”
“It’s fine.”
Twilight grew pink and hesitated. “Right, that’s enough about my past. Could we talk about you instead?”
Sunset licked her lips. “Your new favourite subject, yes.”
Twilight giggled. “Are you familiar with the concepts of connectedness and completeness?”
“Sure.”
Twilight dropped her eyes to the floor, trying on a small grin as she reached out an open palm. On instinct, Sunset took her hand and mirrored the smile. Revelling in the shared touch, Twilight’s smile only grew. “Then mathematically, you would understand what I mean when I say I feel complete when we’re connected?”
“Yeah, I think I see where this is going, and that’s without using my geode.” With a smirk, she added, “But in the interest of mathematical rigour, I would like to see some proof of your proposition– wait.” She blinked, face draining into an empty trance. “Are you about to make a dirty joke about clopen sets?”
Twilight tilted her head. “While connectedness can certainly be characterized by clopen sets, I don’t see why that would be dirty.”
Sunset fiddled with her thumbs. “Maybe something from my world. Never mind.” A quick smile and a squeeze of Twilight’s hand. “Proceed.”
Twilight bit her lip. “Actually, I was going to leave the connected proposition as an exercise for the reader.”
Sunset raised her eyebrow. “Scripted, huh?”
“Sorry, listener. But I do have a hypothetical question for you.”
“Hypothetical? Uh-huh. Yes?”
“Could a human female body be covered in kisses?”
Sunset wagged her eyebrows. “I believe there are songs written about it, yes.”
“But have you ever proven it?”
Sunset leaned in. “I can always find out now.”
Twilight shivered. “No, that won’t be necessary.”
“Ouch.” Sunset pulled back, bringing her hand to her heart. “Here I was thinking this was all a big setup to kiss your girlfriend.”
Twilight eeped awkwardly. “It was? Is? Will be. Just, let me finish?”
“Gladly.”
“There’s a wonderful proof by considering the compactness of the human form.”
Sunset snickered, cupping the right half of her chest in her hand, eliciting a blush from Twilight as she added, “Careful who you go calling compact, Sparky.”
Twilight sputtered. “C– Compactness, the topological notion that for any open cover of a set there exists a finite subcover.”
Sunset winked. “Someone’s been on Wikipedia.”
“Editing.” Twilight rolled her eyes. “Editing Wikipedia.”
Sunset stuck her tongue out in reply, but Twilight only shook her head quickly and continued, “In a Euclidean space, like the one in which we live, a set is compact if and only if it is closed and bounded. And you, Sunset Shimmer, are closed, seeing as your skin is just as much a part of you as the rest of you, and as for boundedness?”
Twilight tilted her head thirty degrees and closed one eye. She formed L shapes with her hands, bringing her thumbs to her index fingers to form a horizontal window she placed in front of her open eye, acting as a frame of reference. “You fit snugly in a bounding rectangle. Hence you are compact.”
Sunset chuckled. “Ah, so I can be covered in finitely many open sets, wonderful.”
“Something like that.” Twilight danced her fingers by her side, in a desperate attempt to stay focused and not correct the mathematical imprecision of her girlfriend’s paraphrasing. “Anyway, the open interior of a kiss is centered around a point with some nonzero size since a person’s lips have nonzero size, so by kissing every single point on a person’s body, they can be quite literally covered in kisses.”
“Infinitely many kisses.”
“Well, yes.”
“Uncountably infinitely many kisses.”
“Well, also, yes, but–”
Sunset pushed her hair off the front of her shoulders to rest snugly on her back, fluffing out her chest and licking her lips. “I’m down. Let’s get started. With uncountably many left there’s no time like the present.”
Twilight rolled her eyes. “While an excellent use of the next eternity” – Twilight squinted, eyes darting up to the corner – “that won’t be necessary.”
Sunset blew some air through closed lips, producing a trill. “Boo.”
Brushing it off, Twilight reasoned, “Since you can be covered by an set of open kisses, by compactness, you can already be covered by a finite number of kisses.” She raised her arm, tilting her bare wrist parallel to her as if to check a watch dramatically. “So if we use the finite subset instead, we can eventually clear through the backlog.”
“I’m in.” Sunset grinned. “How long will it take? I know you’re a pretty fast kisser, but how many kisses are we talking about?”
“How should I know?” Twilight shrugged. “Compactness gives an existence result. After showing its finite, why would the cardinality of the set in question matter?”
“Can’t you just, uh, count them?”
Twilight swatted her hand. “Don’t be ridiculous. Everyone knows mathematicians can’t count.”
“Right.” Sunset sighed. “I guess if my physics degree will mean anything, I better be the one keeping track then whenever we get started.”
Twilight leaned over and kissed Sunset on the forehead. “Does now work for you?”
Sunset kissed Twilight in the same spot in reply, leaning over to whisper into her ear, “One.”
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I want to hope there are clopen sets in this, but it's only T.
10476177
Sincerely hoping there are no closet bronies in my math program who caught on to why I - with an MLP avatar - go out of my way to say "closed and open sets"
Cries in dyscalculia.
Incredible. This is top-tier SciSet flirting and I am 100% down for it.
This was a really cute story! And we could always use more math romance.
Well, possibly, but how rather depends on how we define a kiss and its space, doesn't it? Is the area of the kiss only the area the lips touch, meaning there will be an area in the centre unkissed? Does it include that area, meaning that parts that haven't been directly exposed to the kiss nevertheless count? If so, are any others? Is there an area around the directly-kissed area that nevertheless counts as kissed?
Furthermore, this whole supposition is based on the assumption that a "kiss" has a lifespan of its own that extends past the actual act, which is questionable. After all, a "kiss" technically refers to an action, not an object or a state of being, so one could say that one can only be "covered in kisses" if all points are kissed simultaneously. Which would not only require the centre area to count, but also, since the rest of the various people doing the kissing would otherwise interfere, it would require every set of lips to be disconnected from their owners.
And even if we postulate that kisses linger, we run into the issue of cell life and the Ship of Theseus problem - our skin sheds and regrows naturally over time, so there's a very real possibility that, by the time all kisses are applied, the skin the first were applied to may be gone and require re-kissing, trapping the subjects in a perpetual kiss-cycle. Which, to be honest, these two may not mind all that much, but you get my point.
I mean, don't get me wrong, it's a really cute bit of nerd-flirting, but it does raise some interesting logical and metaphysical questions.
10476180
Also is that math in the story image saying "every open cover has a finite subcover", defining topological compactness?
I like how my approval qualifies as a content warning.
Adorable nerd flirting indeed. I'm glad you posted it for all to see.
10476256
This is one of those rare philosophical issues that can be resolved through sufficient application of lipstick.
10476187
10476258
Yes.
10476256
10476572
Seconding proof-by-lipstick
10476246
10476255
Thank you!
In only finite time, with two pairs of lips! Must be magic!
(Edit) So a Gabriel's Horn could not be covered in finite kisses because it is not bounded.
But what about a fractal with infinite surface area, in a bounded space? (I'm thinking of like a koch snowflake, but as a prism). With infinite surface area, wouldn't that require an infinite number of kisses?
10476734
Consider the closed bounding box (which is compact in R^n), cover that in finitely many kisses, and then you have already covered the snowflake itself. It's ok to end up also kissing the air around the
bacongirlfractal as a coarse approximation, it will still work as a cover.10476746
Ah I see, a spherical shell is fine too. (Reminds me of Gauss's law).
Somewhere in there, I think there's a dirty joke about skintight vinyl body suits.
I didn't know I could dorkily snort so hard. I'm usually more of a goofy giggler.
Back during my Combinatorics college class, Comp Sci, Comp Engineering, and Math students all shared attendance in the course as a possible credit to their degrees. The professor commented once, when a student came in, dropped off their homework at the start of class, and left, "Huh, man, why is it always the Comp Sci majors who do that?". Far back in the lecture seats I called out, "Eh, it's 'cause we think we're better than everyone else." It got some laughs. Comp Sci was by far the easiest major of the other three at our school. It was fun to be brought back to that atmosphere by this.
10666963
I've heard horror stories about all three at my school, so...
As a math major, this is the best math-adjacent fiction I've ever read. The only thing that would've made it better is some sort of pun with Heine-Borel (pronounced very similar to heinie as in butt).
11106155
Shucks, thank you ❤️ I aim to please...
11106725
That... that was truly amazing. I actually had to check if the Heinie-Bare-Elle theorem was actually a theorem in Differential Geometry because I just wasn't honestly sure.
I did not expect such an amazing response to this comment. Thank you.
11106754
10476187 - Well, this is more language than math, so you should be fine? Just think of sets as boxes?
----
I came here from Never the Final Word, and started reading because of the FOME warning! :
----
10476180
Well, as long as they remain in their closets, you have nothing to worry about??
11113394
The FOME warning is there for a reason
And yes they are probably in their closets11106725
This is wonderful. Was totally hoping for a clopen pun. Next, perhaps Twilight can introduce Sunset to the category of Top, and how faithful functors that are injective can lead to embeddings.
There's got to be a pun about the girls and "natural transformations" somewhere, but I'm not smart enough to find it.
11114080
Natural transformations for the girls are nice but group theory is where it's really at, with its homo morphisms.
11114456
How to make homotopy sound even creepier.
I feel like a damn idiot after reading this. especially after reading the comments. wow
11284116
If it helps -- most of the math in the story is obscure and jargon heavy (by non-mathematician standards), but fairly "simple" to learn (not that non-mathematicians have a reason to).