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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 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...
H&K USP 40 compact s&w - Low round count
It comes with a little self-defense ammo (box or two) and 2 mags
Selling to fund my next project. Great Gun!
No Trades, please. Transfer on buyer and can meet within 30 miles of Canby, OR. *FFL of choice is @philcoates (Pioneer Armory), out of Oregon...
In on consignment
Springfield Armory
XD-40 Sub Compact
Two Tone
Comes with original box and 3 magazines
$350 + background check or shipping to your FFL
https://www.highlinefirearms.com/products/springfield-armory-3591-a01d1acc-1fd8-4960-935a-7c2361ce566b
Hello,
I have for sale my p320 x compact. Has the rubber talon grips on it. Comes with a right handed Outside waistband holster and 5 magazines as well
$500
Springfield Armory XDM Elite Compact OSP
Excellent condition, 15 months old with about 800 rounds through it.
Includes everything in the box, five 14 round magazines, five magazine sleeves for use with longer 19 and 20 round magazines (extra magazines not included), OEM #3 optics plate.
Over...
First generation S&W striker fired pistol in 45 acp with three magazines. Comes with roughly 100 rounds of TulAmmo steel case, and 150 in hollow points between frangible and Federal HST
Low round count & pristine condition. It just sits in the safe so it's gotta go. I have a full-size as a bedside gun and I just always shoot that one. The PDP has the best striker trigger of anything for sale right now. It'll come with the 2 factory 15rd mags and the hard case. Irons co-witness...