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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:
Glock 23 Gen 5 .40 S&W, with Trijicon HD night sights, six magazines, and all factory accessories. Less than 200 rounds fired. $420.
Smith and Wesson M&P40 2.0 Compact, 4-inch barrel, with seven magazines, and Ameriglo I-Dot night sights. Fewer than 300 rounds fired. Also with all...
I have a barely used (maybe 50-100 rounds at most) Springfield HellCat Compact HC93198 3” Pistol. Comes with original box, case, 10 round mag, 12 round mag, speed loader and original lock. Cleaned and maintained religiously. Only selling because it never gets fired. $400 or best reasonable...
For sale is a Smith M&P40 2.0 Compact with 4" barrel, five factory magazines, I-Dot night sights mounted, and all factory accessories.
Less than 500 rounds fired.
$420.
I prefer to do transfers at Wooster Armory in Beaverton.
Jim
971.407.8415
Up for sale is a Springfield XD-M Elite Compact OSP 9mm 3.8" with Hex Dragonfly 3.5 moa red dot optic. Near new condition (2 mags through it) with lots of extras. Extras included flared magwell with expanded base plate (14+1), I also added the extended finger plate to both mags to get your whole...
Colt MKIV 80s Series 1991 A1 Compact Model. Ambi trigger, extended slide stop, Wilson Combat fixed sights. One Colt factory 6-shot magazine included. Transfer within the Portland area 600.00
Like new. Never actually took it to the range. Decided to stick with a Vortex LPVO I'm fond of.
Cash only, not interested in trades. Must be willing to meet me near Southwest Scholls Ferry Road and Murray in Beaverton. Thanks.
I'm looking to sell this BRAND NEW Sig Sauer P320 chambered in 9mm. Comes with everything shown. Asking $450.
Buyer pays transfer fee and I prefer to meet in Renton at Pantels Tachtical. I'm open to a more convenient location within reason.
This looks pretty fun, and appears to be a good deal. I've ordered from them before. Service was excellent, and shipping was quick.
https://www.dahlonegaarmory.com/product/sav-axis-ii-300blk-16.1-blk-syn#product_detail
Heckler & Koch HK USP Compact 9mm with LEM trigger, DA/SA trigger, night sights and 4 magazines. Also comes with the two holsters as shown. Used but in good condition.
Asking $899 for everything, or best offer.
HK USP Compact 9mm w/ LEM and Night Sights
HK DA/SA Trigger Parts
HK Magazines x4...
Great condition berett 92x rdo compact comes with everything from factory and x300ua
Want to trade mostly
Looking for 9mm sig p320 packages
9mm Glock packages
Possibly acog or lpvo
For sale is a used but very nice HK USP Compact 45 in .45acp. It is DA/SA with a safety and a decocker. It comes with 5 original HK magazines. Based on what I know about the gun I'd say it has less than 1000 rounds through it.
I am an FFL located in West Linn and I can process the...
Putting some feelers out on selling my Tanfoglio Defiant compact. All steel, with safety,DA/SA trigger. It can take cz75 mags, does come with one OEM tanfoglio mag
Looks like new is over $550, so let's say $350
Looking to sell my x compact
Comes with 2 10rd magazines and a right hand owb holster.
Gun is in great shape low round count. And has the tungsten weight kit installed as well as talon grips.
Asking $500 buyer pays FFL fee.
Open to trades for Glock any caliber just let me know, CZ, S&W...
Recently got bit by the CZ bug so I got to clear out the safe and get some more cash! This is a really handy pistol, looking for a good home.
TLR7
2 fifteen round mags
2 ten round mags
Kydex IWB and kydex OWB holster that both fit the tlr7
Was an M17, and I swapped in a M18 barrel/slide…still not what I’m used to/prefer for a CCW (carried a G19 or a 1911 for years), so I’m selling/trading off for something that fits the bill.
Includes an Olight Baldr light/red laser combo, Black Rhino holster, both flat and curved trigger...