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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.
new take off frame from a 509 Compact MRD
unfired excellent condition.
Tan / FDE in color
Complete frame with all frame parts.
$200 f2f at FFL in Portland, buyer pay transfer
$250 shipped Venmo, cashapp, zelle, money order
add 3% for paypal goods and svcs (no f&f)
Compact M&P2.0 9mm, comes with IWB TXC X1 pro black multicam holster(no light), and Trex arms Ragnarok drop leg holster(with light), streamlight TLR1HL, magazines: X1 10rd X2 15rd X1 17rd, all factory optic plates/palm swells, and if wanted currently has an unfired swamp fox Kraken mounted...
Hi there, I have a basically new Walther PDP for sale. Only pulled out of the packaging. I have upwards of six magazines for it. I believe four of them are extended 17 round. Looking for cash to fix my car. Thank you
Here's one you don't see that often. For sale Springfield Armory Micro Compact 45 ACP 3", 6+1 Rounds for $600. It's just the pistol and (2) 6 round magazines.
Purchased used and only put 100 rounds through it so not sure of the total round count.
No trades just looking to sell to fund another...
I bought this (new) and have run one box of ammo through it with no issues. The model number is #81000331 with a "CC" date code. It's a traditional DA/SA with a safety and decocker with (2) 10-round magazines. Includes:
New (never carried) TT Gunleather, Reinforced Convertible (both OWB/IWB)...
New, unfired. $500
No trades
9mm semi auto handgun. Suppressor sights, optic/red dot ready
It Comes with one 12rd mag and everything else pictured. OEM red dot mounting plate assortment.
Zipper case.
Buyer pays for the transfer at money market pawn in SE or possibly other Portland FFL if you...
Recently purchased this to give the LEM trigger a try. Turns out I still prefer DA/SA, so hoping to sell this to partially fund a different pistol.
Only put a few mags through this. Great condition except for a ding on the front of the trigger guard from the factory. Came with Meprolight Tru...
Selling off my S&W m2.0 compact magazines. I have 8 total and each are equipped with Floyds custom base pads making them 17 rounds total. All of these are under a year old.
Brand news these would cost $70 each. I am asking $299 obo for all of them.
No trades please.
Excellent condition, maybe 3 mags through it. Comes with: Case, Gun, (1)9-Round Magazine, Manual.
Local pick up preferred. Buyer pays for shipping (if applicable), transfer & background. I accept all payment methods - please add fees if using goods and services.
Preferred...
Selling a CZ P10 Compact chambered in 9mm, it is in excellent condition and comes with the factory case and 2 mags. . Asking $460
I am not able to take trades on this item, cash deals only (no payment apps) and the transfer is to be completed in Hillsboro at my ffl. please make all contact...
Selling my two S&W M&P 2.0 9mm (Full size and compact 4")
Both of these have been upgraded, but include all original parts (Base pads, palm swells, sights, and triggers)
The full size comes with the following:
Trijicon night sights
Apex flat trigger
Floyds custom magwell
Floyds Plus 6 base pad...
Selling my concealed carry handgun. Lots of mods on this and selling all of them with the gun.
Well taken care of and very clean.
All original parts included (sights, trigger, basepads)
model: Smith and wesson m&p 2.0 4" 9mm compact 15 round
Add ons:
Floyds Custom magwell (msrp $75)...
S&W M&P 2.0 Compact 4.0" with a manual safety, 15rd flush fit capacity. Has shot flawlessly, maybe 200rds through it.
Includes original box and contents. Can include 1-2 mags for extra in a legal state but I'd rather hold onto them due to being stuck in Washington state.
Also selling...
Looking to pick up any CZ mags that work in a 75 compact. Willing to drive reasonable distances to meet up. I’ve got some 5.56 I’d be willing to trade or just buy outright.