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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.
Hey Y'all! I'm selling one of my Shadow 2 compacts! This one is fully built by CGW to be an absolute beast. It has a SAO trigger, Vytal grip in emerald green, 4 magazines, a black kydex holster, and all original components (including mag release, baseplates, and grips)! It also comes with the...
For Sale is a NEW P320 9mm Compact/Carry Slide with NEW Siglite Night Sights recently installed. This Slide is a NEW Old Stock take off from a P320 Compact. The slide does not have an optics cut. For those who like their iron sights, this is a great one to have.
If you have any questions...
Clearing out some pistols I haven't shot this is a never fired Springfield armory XDM ELITE Compact 3.8 barrel COMES WITH 2 mags and factory soft case
may trade for seekins precision ar parts ,carbon fiber barrel , 15 power binoculars 14k scrap gold
This gun sells new for 450 I'm asking 325 it comes with 2 mags. I fired 50 rounds with no problems. I can meet in the Salem area for transfer and buyer to pay all FFL fees.
DA/sa. Looking to trade or sell, went out and shot a few boxes through it, just doesn't fit my hand right. Other than that it's brand new in the box. Will trade for other 9mm of similar value, or the right deal. Will sell but firm on price.
I have for sale a Smith & Wesson Model 3953, a 9mm compact single-stack 3rd generation Smith & Wesson pistol. This pistol has an aluminum-alloy frame and a steel slide. The manual-of-arms is DAO, with no safety/decocker. This pistol has iron anti-snag Novak style Meprolight green tritium night...
I have for sale a Smith & Wesson Model 3914, a 9mm compact single-stack 3rd generation Smith & Wesson pistol. This pistol has an aluminum-alloy frame and a steel slide. The manual-of-arms is TDA (DA/SA) with slide mounted safety/decocker. This has an ambidextrous decocker lever, a low-profile...
Lowest price ever on the price history chart. Ordered one for my .22 rifle as I'm tired of using iron sights. Curious if anyone has tried this optic before. I figure it's in the same tier as the cheap Bushnell ones. For a larger caliber you may want to use something a bit robust like an Aimpoint...
Like new CZ P10C
2-15rd
1-21rd magazines
Factory accessories
$330 FIRM cash price
$350 trade value
Trade for:
Top of the list trade is A2 carry handle upper plus cash (stripped upper works)
1-6x24 vortex scope with mount preferred
S&W revolver or semiautomatic metal frame pistols, 39 or 59...
I have a Beretta APX one carry compact.
Two magazines 13 rounders
All three back straps
Original Tupperware
Buyer pays transfer
Open to all trades
Currently looking for
9 mm ammo
300 blackout ammo
45 ACP ammo
9 mm handguns
40 caliber handguns
Any 22 caliber
AR style rifles
Excellent condition just need to sell and thin the heard as I have another project I need the money for, comes with Vedder holster and four 15 round magazines and case. I will post this week only for the price of $350 and the buyer will pay for the background check. I would prefer to do the...
Excellent condition P320 with four magazines (15) rounders. Downsizing and looking to get a HF Ham radio for my office area. I not willing to change the price and the buyer or person trading if they have a HF Ham Radio will pay the background fees, I would also prefer to sell in Salem and do...
Picked this up as an understudy to my Shield but I haven't ended up shooting it that much
I got it used so no idea on round count, I have heard that some of them preferer HV ammo but I took it out the other day and ran 2 mags each of CCI SV and Federal Champion and it ran fine
Comes with
4...
Got a brand new never fired Walther Pdp 4" compact 9mm for sale. Comes with everything from factory including 3 magazines. Just bought a compact and full size and decided to keep the full size.
Interested in trades.
- AR15 (A2)
-Glock in 9mm
- Single action revolvers
- Ruger IV 22/45
impulse bought a Walther PDP 4" 9mm with Ameriglo haven. Just looking to see what trades are out there. Only interested in handguns and rifles (not hunting rifle). The handgun is brand new and never been fired just racked a few times..