//------------------------------//
// Fillyfoolish's Quod Erat Duvet-Strandum (Fillyfoolish's "Topology") 
// Story: Never the Final Word (Vol. 2) 
// by FanOfMostEverything 
//------------------------------//

“You fit snugly in a bounding rectangle. Hence you are compact.”

Sunset quirked her eyebrow. "Remind me, what does fitting into a bounded rectangle have to do with open covers?"

Twilight frowned. "You never learned about the Heine-Borel theorem?"

Sunset grinned deviously and reached her hand under Twilight's sweatpants, cupping her bottom cheek and squeezing. "That theorem?"

"N--No," Twilight said behind a pronounced blush, quick breaths, and a faint smile. "That's the Heinie-Bare-Elle Theorem from differential geometry." She swallowed. "Completely different."

Sunset retreated her hand and planted a little kiss on Twilight's ruddy cheek. "My bad. Do you mind teaching me about this Heinie-Bare-Elle Theorem?" She flashed a smirk. "Please present a rigorous proof."

Twilight skipped a breath. "....Tonight? My place?"

Sunset wagged her eyebrows, accepting the invitation by bringing her fingers to her lips and blowing a kiss.

Twilight hid her joy behind her hand as she pushed up her glasses. "The Heine-Borel Theorem, on the other hand, states that a subset of Euclidean space is compact if and only if it is closed and bound. The proof that closed and bounded implies compact is by contradiction. Let C be an open cover of a closed and bounded set S such that C does not have a finite subcover..."

Sunset curled a lock of her hair behind her ears as she listened, more to the familiar sound of her girlfriend's voice than to the words modulated over that voice, and smiled gently.

"...In conclusion, because you are closed and bounded, you are compact."

Sunset chuckled. “Ah, so I can be covered in finitely many open sets, wonderful.”