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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 my Smith & Wesson M&P9 2.0 Compact 3.6". Less than 300 rounds fired through it. Comes with 4 15 -round mags and 2 17-round mags. The box is around somewhere but I don't know where so the grip size that's on it is ML (Medium Large) and may be the only size it comes with. No trades.
Hello all, looking to trade this CZ 75 compact I recently acquired which has been upgraded by Cajun Gun Works. I’m looking to replace it so soon because shortly after I got it from another member I was gifted another 9mm that fills the same role. It’s got a fantastic trigger and the G10 grips...
CZ 75 Compact P01. 9mm. 14+1. Decker. Two mags. Case + paperwork. Original owner. Like new condition. Maybe 150 rounds shot.
Gun was top rated 9mm in the world for reliability. Great hand feel. Minimal kick. All Metal. $595 includes de Santis OWB leather holster.
Will do trade+ $ for 6.5...
Looking for fellow PX4 owners that may be interested in trading some magazines? I have four compact mags 15rd, but I need the full size 17rd. Or, I would like to buy full size mags.
New Pelican-brand foam, rubber seal perfectly in tact. Internal dimensions are 8.375" wide x 5.875" tall x 3.25" deep. Not interested in trades right now.
Comes with two 15-round magazines, SIG x-ray night sights, and a compact frame size. Comes in an original case with paperwork, gun lock, and chamber flag. Very few rounds were fired through it, less than 500.
Got one last time and it works well on my .22 plinker. Great price per the price history
https://www.preppingdeals.net/deals/amazon-crimson-trace-cts-25-compact-red-dot-sight
Clearing out the safe. 40 cal, has original paperwork and plastic case. Comes with 4.5 boxs of ammo. I have put 25 rounds through it. You pay transfer fee's
I have a nearly new (about 200 rounds through it) Beretta PX4 Compact Carry in 9mm for sale. This is the version that they did in cooperation with Langdon Tactical. It does not have the trigger action job, but it is still a VERY good and smooth DA/SA trigger system. I'm only selling because I...
Ruger American Compact 308 - 18in with Vortex Crossfire 3-9x40 and a level bubble (non-threaded).
I bought it 2 years ago but haven't had time to practice with, only had around 50 rds through the rifle.
The rifle has some cosmetic damage only on the right side of the barrel due to a leakage of a...
Excellent like new in box with decal and warranty card. Red laser rubber overmold grips for 1911 officer or compact model.
$175.00 or trade for Wilson or Chip McCormick 1911 mags in 45ACP or sub sonic 9mm, 300bo, 22lr ammo.
Looking for Glock oem or aftermarket slide/uppers for compact gen 3 frame I would prefer a upper chambered in 357 Sig or 40 S&W but would also be interested in one chambered for 9mm or other caliber(45GAP Ect) let me know what you have and asking price looking to buy or trade asap. Really...
It comes with all the factory stuff + 2 X 17 round mags and 1 X 21 round mag. Just got it broke in for you...500 rounds through it. Model # JAXA1C915FO
I know this is a long shot so I'm trying to sell some things to cover a dog surgery that was a lot more than anticipated.
Up for consideration is an Gen2 compact acog 1.5x16s. Never mounted. This thing is pretty cool and I hate to sell but the dog comes first.
Looking for 700 or obo.
Yes, FREE. I'd humbly suggest you donate something to NWF if you pick this up. I got this in a bundle purchase and I'm right handed. No brand on the Kydex holster. It is LEFT handed. Meet my at the Fisherman's off the I-205 near Oregon City/Gladstone. I'll email a photo if anyone wants this.
I...
New xdm elite osp 9mm compact. This is the osp model slide is cut out to take red dot or reflex sight. Gun is new and never been fired, I ordered it but I prefer single stack for carrying. Asking $350 and do transfer at a cut above pawn in beaverton. Buyer pay transfer fees.