NNNNNNNNNNNNNNNOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. IT'S HIM. HE'S BACK. THE HOLEINMYHEART GUY. THE TWO JIGGLYPUFFS ON THE FRONT PAGE FAGGOT. GET THE GUN. SOMEONE, GET THE GUN. SHOOT HIS BRAINS OUT AND SPLATTER THE WALL WITH PINK. WE CAN'T HAVE THIS DEGENERACY ON NEOCITIES. QUICK: REPLACE THIS GODAWFUL 1GB OF SPACE WITH EROTIC FURRY FANFICTION. FUCK, FUCK, FUCK.

Ha, ha, ha, hAAAAAAAAH. Welcome BACK, you fuckers. I'm a CLEAN, COCKY pink and you're blemished with blue. How's the new year going for you guys. Oh, feeling blue... eh... "OMG, there's VIOLENCE ON THE CAPITOL. 2021 WASN'T SUPPOSED TO BE LIKE THIS!?!?!?" OMG, shut your FUCKING MOUTH. FUCK YOU. GO FUCK YOURSELF. No, seriously: Stop freaking out and watch some idolm@ster PVs.

Hartshorned?!?!?! What does that mean? A heart to be shorned? A mongrel at math attempts to peep into forbidden knowledge? A lonely soul, publishing some contents of his heart in a oblique manner? Orrrrrrrrrrrrr, DUN DUNNA DUN DUN DUN!!!!! IS IT FAILURE? The smell of SLOWNESS. The drag through the exercises at a pace that won't have me done with em in a reasonable amount of time. The DEMONSTRATION OF INCOMPETENCE. Ah, yes. HARTSHORNED. H-A-R-T-S-H-O-R-N-E-D. Heart of my hearts. In the morning, she was Harty... What?

Last time, and it's been a while, we did 1.7c. Now, if you've been reading lclosely, you'd know that there IS a 1.7d. I can't trick you, reader: It exists. So why have we docked at the port of 1.10 to begin the new year? Well, you'll see when I get around to 1.7d. But in the spirit of NEW HORIZONS, but still not quite new enough to start a brand new section, bugger, here's 1.10a.

We wanna show:

(1)

So suppose

(2)

is a chain of irreducible closed subsets of Y . NEW MEME: I hate writing and typing out "irreducible closed subset", so my new term for this is going to be "irrc". IS THAT CLEAR? "AYE" Thanks, sailor. So we have a chain of irrc's of Y . And BASICALLY, we want to create an at least equal length irrcs chain of X. Savvy?

At first I tried to use the subspace topology of X directly, say using C_i = D_i ∩Y where D_i is closed in X. But D_i ain't necessarily IRREDUCIBLE, like we need, so this was a dead end to me.

So then I decided to phuck around with closures. Thanks to an old friend 1.6 (it was measly 4 exercises ago, so it shouldn't really be considered "old", but, yknow, at the pace I'm going...) I know that


C_i irreducible cl_X(C_i) irreducible.
(cl_X means "closure in X")
NOW, since C_i is closed in Y already,

clY (Ci)	= Ci			(3)
Y ∩ clX(Ci)	= Ci	with a little help from here		(4)

BUT,

Ci	⊊ Ci+1	(5)
Y ∩ Ci	⊊ Y ∩ Ci+1	(6)

BUT ALSO,

Ci	⊊ Ci+1			(7)
clX(Ci)	⊂ clX(Ci+1)	Closure preserves inclusion... I think		(8)

So combining (6) and (8) yields

(9)

Sooo,

(10)

is an irrc chain in X. I.e. Any irrc chain in Y has a corresponding equal length irrc chain in X, so the maximal chain in X is at least as long as Y 's. I.e. dimY ≤ dimX.

DONE. DID YOU ENJOY THAT, COCKSUCKERS? Announcing my return with the Hartshorned blog? Badly formatted equations? Some of the pictures are too tiny? Hahahaha HA, might as well get rid of the rest of the site. We'll see, we'll see.