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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.
M&P 2.0 Compact 9mm (4")
This one is pretty clean. Under 100rds through it. Purchased this a few years ago and it's been in the back of the safe since. Very good condition.
2 - 15rd mags
1 - 17rd mag with sleeve(s)
Comes with everything that was in the original box. Plus the extra 17rd mag...
Fo Sale is a New in the Box Smith & Wesson M&P9 Compact 2.0 with 4” barrel. This was mfg in 2021 and comes with box, manual and (3)-15rd mags. This pistol is optic ready, has suppressor sights and Apex threaded barrel (have factory barrel too). Mounted is a Trijicon RMR Type 2 Red Dot (3.25)...
Hello,
I have a Taurus G3C compact pistol in 9mm available for sale. I purchased this from a fellow forum member a while back and have only put around 50 rounds through it. It’s in excellent, nearly new condition and comes with original box, lock, documents and two 12 round factory magazines.
I...
I’ve got 3 mags for a compact 1911 that I ordered accidentally, didn’t realize it till after I threw away the packaging.
They were $35 each when I got them and haven’t been used once so I’d like $25 per mag.
Brand new unused just don’t have the packaging anymore otherwise I’d send them back
I’d...
Like new OZ9C. Around 350 rounds shot.
Why get a basic glock and then pay the cost of the pistol to swap out the trigger and get the slide milled to still look basic as f*ck?
Comes with original documents. WA residents will get one diet 10rd glock mag and everyone else where applicable two...
Like new Brugger and Thomet GHM9-G Compact. This has the Glock lower and accepts 33 round mags. $1400 with transfer fees on buyer. Have extra mags and factory B&T stock available for additional $. Thanks for looking.
Thinning out the safe a little and transitioning to Glock for the most part.
320 compact has had less than 100 rds through it, 2 15 round mags included, and factory case. Will post picture when I get home.
Trade options:
Glock 17, 19, 43
Interesting .22 LR pistols
9mm target ammo
Steel...
I am selling my P320 setup minus the FCU. It includes the following:
-Sig P320 AXG Carry grip module with factory G10 grips and upgraded hex grip screws
-Sig P320 Compact slide with all factory internals, 3.9” barrel, recoil spring, factory suppressor height night sights, optic cut for RMR...
selling my p320 compact as a kit for anyone interested in just the kit and not the FCU. Just the cost of the mags make this a steal!
-slide
-barrel and recoil spring
-grip module
-tlr7
-two 15 round mags
-two 10 round mags
-kydex iwb holster
-kydex owb holster
Not interested in trades...
I need to get a USP Compact .45 IWB Holster. Any ideas who to order from? Looking for Kydex preferably but not oppose to leather. Any information is appreciated thanks!
Unfired walther pdp 4 inch compact with surefire x300ub comes with everything from factory
600$ without light
800$ with light
Trade +/- cash
Quality ar uppers
M1 garand
1903 Springfield
K98 Mauser
Possibly other
For Sale is a New in the Box S&W M&P9 compact in 9mm. This has a 4” barrel and was mfg in 2021. Optic ready, Apex threaded barrel, box, paperwork, (3)-15rd mags and back straps. Also has suppressor sights and comes with factory 4” barrel. Also can come with Trijicon RMR Red Dot Type 2 3.25...
For Sale is a New in the Box S&W M&P9 compact in 9mm. This has a 4” barrel and was mfg in 2021. Optic ready, Apex threaded barrel, box, paperwork, (3)-15rd mags and back straps. Also has suppressor sights and comes with factory 4” barrel. Also can come with Trijicon RMR Red Dot Type 2 3.25...
Looking to see if anybody is selling USPC 9 mags, has a nice HK USP compact 9mm Holster or other accessories for the USP compact 9mm send me a message with what you have and your price.
Hello,
I have a CZ P01 Omega in great condition that I would like to trade for a Beretta PX4 Compact or Compact Carry.
The P01 is in great condition and has the MCarbo trigger upgrade.
I am looking for the PX4 Compact Carry (the Langdon Tactical version.) plus some money on my end.
Trades...