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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.
I have for sale today is a FN 509 Compact with 5 mags, 1x12,1x15,3x24 never fired
Not interested in trades at the moment
If you meet me at my FFL, there will be no transfer fees for the buyer
I've got a raw SCT 19 Compact Frame in Outdoor Green with an included Strike S80 magwell. This is a serialized part that needs an FFL transfer.
This frame can be finished with PSA Dagger or any other Glock 19 gen 3 compatible parts.
I bought this, held it and didn't like how the trigger...
Model: Dagger Compact (G19 size)
Optic: Gideon Alpha (RMR Compatible)
I've got a 2-tone PSA Dagger Compact (Compatible with Gen3 Glock 19 components). The dagger has a black frame with a SW2 RMR slide in sniper green and a threaded barrel.
This comes with an installed and zeroed Gideon...
Excellent condition. Orginal owner with less than 400 rounds through it flawless. Comes with Holosun HE508T-GR optic titanium multiretical 2MOA and/or 32 MOA. 2 17 rounds magazines and 2 15 round magazines. Only selling because I upgraded to the Pro version to start competition shooting...
Walther pdp 5" compact
7-18 round mags
2-15 round mags
Rmr plate
Walther case
$650
Will sell the gun, plate and 2-15 round mags for $550.
Tlr1-hl can be included for $100 extra
Hey Gang,
I just picked this pistol up a couple weeks ago for an awesome dude on this forum. I put less than 50rds thru it last weekend and I do like it a lot, but I don't plan to replace my current CC piece. And I picked this up mostly out of curiosity since I had seriously considered one...
I bought this a month ago and have fired 150 -200 shots through it. Unfortunately, the way my finger contacts the trigger doesn't work well for me. My loss is your gain.
Condition is great, as pictures show. Includes:
1-12 Round Magazine
1-15 Round Magazine
3 grip back straps
magazine loader...
Up for sale, Southwest Washington, Vancouver area, Clark County area.
Trijicon ACOG. 2 x 20 TA47-C-400341 Compact. Green illuminated crosshair. Always on. Crystal clear and wide field of view. Bright fiber optic, and Tritium illumination. With Q-Loc mount. Ready to slap on an AR or similar...
Selling / trading a Sig p320 Compact. Comes with (4) 21 round mags, (1) 15 round mag, and (1) 10 round mag. Magazines alone would have 200 trade value.
Looking to trade for pistol optic (eps, rmr, sro), tlr 7 sub, apex trigger set for mp 2.0 compact, threaded barrel Added cash on either side...
Very nice Weaver R4 made in Japan...from the 2000s...matte with duplex reticle...From what I have read they were a highly regarded optic...Nice and compact...
Minor marks from weaver rings....
Asking 115 shipped/insured.....pm me...
It's in 45 ACP. I'm the original owner and haven't shot it in years, so it needs a good home. It has a 4" barrel and the frame is all steel with a gray lower and black slide. Fluted chamber and barrel. Ambi safety with carry cuts. Original bag and paperwork. One 7 round mag and two 8 round mags...
Like new condition, CZ Compact, steel frame 9mm. New 2019. Full inhouse Cajun Gunworks treatment: (w/ paperwork)
Pro package, race hammer, reduced reach trigger
10x bushing
SS guide rod
Optics cut
Slide cut for weight reduction to offset optic
Thin CZ aluminum grips
Holosun 507 green dot, w/...
Up for sale is a 3.8 XDM Elite 9mm compact with Hex Dragonfly red dot. Only reason I am selling is because I have 2. I bought this on a whim and was going to set it up the same as my other one but decided not to. I did take it to the range and put one box of 50 rounds through it with zero...
For Sale, Canik TP9 Elite 9mm, Sub-compact, Optics Ready, one 12Rd mag, one 15Rd mag, includes $60.00-Taylor Freelance brass backstrap, comes with Factory case and accessories. Bought this gun brand new about 4 years ago for $430, my son really liked the gun, so I gave it to him for his 21st...
Looking for a compact Glock or P3 65 or similar but really interested in just anything so let me know
Trying to trade my Aero upper or holosun EPS green dot
Radian raptor cc
Aero bcg
Mid west hand guard
Mbus flip up irons
Daniel defense foregrip
Rail covers
RMSc footprint. Comes with lots of screws for different slides, I believe 7 sets. Shake awake, up to 50k hours of battery life. This one will include the hood for added protection. It is a red dot, no green.
Will take trades in ammo, 308, 5.56/223, 300 BLK, 9mm.
I'm looking for a metal framed DA/SA in 9mm or .45 ACP, compact or full-size, either OEM or cheap-but-functional clones. No sub- or micro-compacts.
Examples:
Browning/FN Hi Power
Colt 1911
CZ P-01
Beretta M9 or 92FS with straight dust cover
No M9A1/A3/A4
This is the only example where I'm not...