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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.
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..
2 Tone
Rosewood Grips
Factory Case
2 Magazines
I'm also throwing in a 20 round box plus 11 lose rounds of Sig Sauer Elite Performance ammo and 50 rounds of Browning ammo.
I live in Snoqualmie. If you have a preferred FFL I will travel up to 50 miles to meet. Buyer pays transfer fee.
Psa dagger compact (Glock 19 size) in sniper green with a bronze barrel
less than 100 rounds down the pipe
just need to thin the safe out and get some cash
absolutely nothing wrong with it, grip tape is just preference and can be removed
will include the PSA dagger small “range bag” that comes...
PSA Dagger Compact (G19 clone): 9MM Pistol with Extreme Carry Cut Slide with Night Sights, Flat Dark Earth
-Flared magwell/magazine adapter sleeve- accommodates a G17 magazine in the Dagger Compact and fills the gap between the magazine and grip
-Tear drop extended mag release
-Upgraded night...
Looking to sell my Springfield Range Officer Compact 45 ACP. It has been a great gun but mainly just sits in the safe at this point. Generally carry my full size 1911 hunting and p365 as EDC now. Probably around 400-500 rounds. Will include the 1911 ultra tool for takedown, and two don hume...
For Sale is my brand NEW Factory Take off P320 9mm Compact Barrel and RSA (Recoil Spring Assembly). It has a 3.9" length with no LCI. This barrel and RSA came from a NEW compact P320. If you have any questions, please message me.
$145 Cash for local buyer. $150 Shipped from Oregon. I Accept...
Custom no crappy regulator(must wait 5 seconds between shots for reg to function). No reg. means more power, bigger tank more air and less weight. This gun has NO business having a regulator. 10; inch barrel semi auto made for plinking/varmint control. It will never be a bench gun. No reg also...
Up for sale is my G19 clone. It is a PSA Dagger complete lower in sniper green with LFA upper. It comes with one 15rd Pmag. I will add pics when I have time later.
$275
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Holosun HE507K-GR-X2 Compact Multi-Reticle Green Circle Dot Open Sight w/ Shake Awake
I have 2x brand new units for sale. They haven't even been removed from the box. Bought these a few months back and ended up going with another setup.
Asking $270 each, price is...
Well I've been debating selling this for a couple years now and I've finally decided to see if I could find a home for it. I just don't use it much. I've put maybe a couple thousand rounds through it at most. Almost all were subsonic. It has been completely reliable with both subs and supers...
Ruger American Compact 308 - 18in. Great for backcountry hunting without breaking a budget.
Vortex Crossfire 3-9x40 with a level bubble.
I am throwing in about 55 rds of different type of hunting ammo for you to choose.
I just want to be up front that the rifle has some cosmetic damage only...
Selling a Glock 19 2020 Gen 5, asking $475.00. Brought brand new Date 03-04-2020. I never fired this gun nor took it out of the case. No scratches and all complete. Background check goes to the buyers responsibility. BGC to be done at Tigard Pawn 4 More. Send me a message if interested.
Because I like to sip bourbon and order things I didn’t pay attention and bought the wrong magwell. Unused. Cash or trade for similar gen 5 Glock 17 magwell. Maybe as part of a trade for a surefire x300 1000.
Lightly used Beretta PX4 storm compact in 9mm. Will come with 2 mags and the original box (users manual etc). Gun has barely been used and is in great shape!
New Years Sale Pricing!! Price includes shipping AND tax ($429.99+tax=$457.94). Please reach out via PM with any questions or offers...
For Sale is my lightly used 3.9" Compact/Carry Sig Sauer P320 Spectre Slide Assembly! Comes complete with NEW barrel and RSA in EXCELLENT Condition!!! This has the updated Optics cut for Romeo1 Pro/Romeo2/DPP & RMR/Holosun footprints. Also has a set of the XRay3 Tall Sights. I only used this...