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compact
In mathematics, specifically general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (containing all its limit points) and bounded (having all its points lie within some fixed distance of each other).
Examples of compact spaces include a closed real interval, a union of a finite number of closed intervals, a rectangle, or a finite set of points. This notion is defined for more general topological spaces in various ways, which are usually equivalent in Euclidean space but may be inequivalent in other spaces.
One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space.
The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded.
Thus, if one chooses an infinite number of points in the closed unit interval [0, 1], some of those points will get arbitrarily close to some real number in that space.
For instance, some of the numbers in the sequence 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, … accumulate to 0 (while others accumulate to 1).
The same set of points would not accumulate to any point of the open unit interval (0, 1), so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact since it is not bounded.
For example, considering
R
1
{\displaystyle \mathbb {R} ^{1}}
, the entire real number line, the sequence of points 0, 1, 2, 3, …, has no subsequence that converges to any real number.
Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion—and the standard definition of the unqualified term compactness—is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally—that is, in a neighborhood of each point—into corresponding statements that hold throughout the space, and many theorems are of this character.
The term compact set is sometimes used as a synonym for compact space, but often refers to a compact subspace of a topological space as well.
For Sale is a like NEW Sig P320 9mm Compact grip module and holster. This is a take off from a New but Old stock pistol.
$30 Cash for local Buyer. $35 Shipped from Oregon. I Accept PayPal Friends & Family, Venmo or Zelle for Shipped items.
For Sale is a Brand NEW Old Stock take off Sig Sauer P320 9mm Complete Compact Slide Assembly. This slide has contrast sights, barrel, RSA and is cut for the current trigger disconnector upgrade. The pictures speak for themselves. If you have any questions, please PM me.
$350 $320 Cash for...
Have a new/unfired walther pdp compact up for grabs, comes with two mags. Light is sold/not included
Trade interest
3rd gen Tacoma overland parts
Hi power or clone
Steele frame s&w 9mm
Gp100/686
P365 x macro
Lpvo
Magnifier
Shoot offers
Up for sale, Clark County WA, Vancouver area.
Meet up at Hazel Dell pawn shop for transfer.
Ruger American Rimfire Compact. .22 WMR / .22 Mag. Fiber optic iron sights.
Package deal includes six magazines. And 100 rounds of ammo. With option to include a scope. Very handy walking varmint gun...
Selling a barely used Walther PDP comes with 4 15 rounder mags and number one optic plate for your optic. For sale or trade let me know what you got? Worst I can say is no thanks!
Up for sale. SW WA, Vancouver area, Clark County. Meet up at Hazel Dell pawn shop for transfer.
Ruger American Rimfire Compact. .22 Magnum, .22 WMR.
Package deal includes 7 magazines. Weaver scope bases, and 100 rounds of ammo. Stock butt piece is interchangeable. It has the compact one on it...
Like new Sig Sauer P320 Compact / Carry Pro cut slide.
Just swap over you compact or carry barrel and recoil spring and you’re good to go.
No trades please.
new psa dagger 2 tone sw1 compact with 1 15rds mag and carrier bag and vortex venom red dot
ita new less 50rds on it
open for trade for
1911 45acp or anything you can offer
shoot me offer
$500
Can’t justify the is gun is sitting, selling a barely used Walther PDD compact comes with 4 mags, number 1 optic plate, for sale or trade 550
One of the best striker fired triggers out there.
Hey all, up for sale today is my GHM9 Compact w/ some extra goodies. I was building this pistol with intent to Form 1 it and turn it into an SBR. However, I am now going another route.
The pistol will come with original case, literature, 3x B&T magazines, B&T sling, and the following optional...
Used HK USP compact 45 light holster wear Da/SA with decock and safety
3 dot night sights are still good
4 magazines some have mag extensions
Private message me if you have a trade offer mainly looking for pistols with optics or a rifle.
$680 cash buyer pays transfer fee or background check
Purchased everything new for this build to train my son for 9mm. He has moved on to 9mm and this is no longer needed. Exactly 200 rounds through pistol. Here's what I paid…
TX22 Compact - $330
Holoson 507k GR - $320
Streamlight TLR7a - $130
2 Mag Extensions - $30
3 Xtra Mags - $65
TOTAL : $875...
Like new condition. I think the CZ P-10 is up there with the Glock 19 in terms of the absolute best compact 9mm on the market. This Gen2 model comes with a black serrated rear sight and front night sight with a bright orange ring. The glove-like grip alone makes this gun legendary. Selling due...
For Sale is a lightly used Sig P320 Compact Silver 9mm Threaded Barrel by Armory Craft! It has the 1/2x28 Thread pitch. Barrel functioned flawlessly. This fits the 3.9" Compact Slides.
$110 Cash for local buyer. $120 Shipped to states where threaded barrels are legal. I Accept Discreet PayPal...
Got these in trade a while ago and they just sit in the safe.
The Compact has a brand new slide with 2 mags through it. Haven't taken the full size out yet.
PSA Dagger Full Size (Unknown rd count) - $500
Stippled Grip
Gold threaded barrel
Compensator
Vortex Venom Red dot
Auclaytec light
2 spare...
Selling my Tanfoglio Defiant Compact, all steel. Imported by IFG, not EAA. Comes with 1 mag (uses cz75 compact mags) and case. I'm asking $340, no shipping so Oregon only.
9mm B&T GHM9-G Compact. Glock lower; includes 1 33 round magazine and factory hard case. Additional 33 round mags and folding B&T stock available for extra $. $1250 with transfer fees on the buyer. Will consider partial trade for a .45 Sig P220 or HK USP. Please communicate via PM. Thanks for...