Search the forum,

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.

View More On Wikipedia.org
  1. G

    Saniflo Compact toilet-water flows into pan

    Hi there I fitted a Sanicompact 43 about10 years ago. A few years ago i noticed that it would intermittently activate in the middle of the night so I fitted a simple pull cord swith to turn it off at night About a month ago I recently had a blockage so pulled it apart and cleaned everything...
  2. Greenstar123

    Greenstar 30si compact filling link

    Hello all, I want to increase the system pressure to 1.5 bar. Following the user manual (found one online) and some YT videos the filling link should be clearly visible there, but I still cannot find the filling link on my Greenstar 30si compact, or at least nothing there looks like it. It...
  3. A

    How easy is it to uninstall a compact dishwasher yourself?

    Easy if you're a plumber I guess, but if you have no plumbing knowledge? Its a Bosch, nearly ten years old I think, its door won't shut, so it needs to be removed. They're a few videos that supposedly explain how to do it, but none for countertop dishwashers. I only saw one mention what you...
  4. B

    Saniflo Compact 43 toilet

    My toilet has been randomly doing a small flush. I've had a visit from a Saniflo engineer who has checked the unit. There is no blockage and it is flushing away correctly. He has replaced the non return valve and the solenoid valve but this has not solved the problem. There is a very slow...
  5. P

    Worcester greenstar compact 30Ri burner.

    My Worcester greenstar compact burner green light is only firing up for 5 seconds then cutting out. Anyone got any ideas before I call an engineer out. Incase it’s something simple.
  6. H

    URGENT Sani Compact gives off gas odor

    Sanicompact C4 SFA Saniflo 120 V Sanicompact Recently installed this unit and gas odor is strong when flushing, have replaced part F but odor persists. What can be done to fix this?
  7. M

    Worcester compact service tool

    Hi all. Got a few of these boilers coming up to 4yrs old and want to know if anyone has used something other than the recommended service tool which costs £57?
  8. K

    Vokera compact 29a pressure almost 0

    Hey, I noticed today that the pressure has dropped to almost 0 in my vokera compact 29a boiler. There is a flexi pipe just below with 2 black taps and usually turning those will increase the pressure but it’s not working this time. I have a picture attached. The tap on the left let’s water...
  9. C

    Glowworm Compact 24c combi

    hey guys, I have an issue with a combi, its brand new (4 months or so) and it will not stay fired up for heating, when I originally went the boiler had no thermostat so I fitted a stat thinking it would sort it, yet it calls for heat, warms up to required flow temp and return warms up and then...
  10. T

    Boiler swap. INTEC 40GS to Worcester Greenstar 32CDi Compact Combi

    Dear Forum members, Hope that you are all doing well! We are thinking of moving our existing boiler from its cupboard to a bathroom corner to free up some space A local specialist has recommended that we install a smaller boiler, so that it fits into the bathroom nicely My goal is to ensure...
  11. S

    Biasi Riva Compact HE MK2 32 SM/C

    Hi Guys, Have a Biasi Riva Compact HE MK2 32 SM/C. Boiler is not igniting for heating or ho water. The operating light is flashing green however when a hot tap or heating is turned on the light keeps flashing green and boiler is not going through ignition sequence. Any ideas, gas supply is...
  12. B

    Compact format water softener

    I rent an apartment in Santiago, Chile where the water is very hard. For the last few years, I've been using the building's laundry on the ground floor but this costs a fortune over the long run and is frankly a ballache when you're on the 12th floor. A colossal waste of time. Not to mention the...
  13. I

    Boiler Switching off Heatline compact s24

    Hi all, Well i think this is my first thread ,and you've guessed i have a problem so ill get straight into it. Decided to install some Fernox F3 into my system as i have been cold spots on some radiators,ive used this before and its always done the trick. Anyway i dropped all the water in the...
  14. Russell3162001

    Old Heatstore Compact

    Ive got an old Heatstore compact - removed the old screw but the front cover won't move? Any ideas on how to free it (Stock picture - I guess the shower is somewhere between 20-30 years old?) Close to using a hammer to do it!
  15. G

    Error 385 on Baxi Megaflow 2 GA compact

    Hi, I am getting error code 385 on my boiler display which online states 'low voltage'. Any idea on what I can try? Someone suggested replacing the fuse but no idea where to locate it, can anyone help? Thanks
  16. G

    Main eco Compact 30kw question on rear pipework.

    Ive heard its possible to get the pipework up the back of this appliance by removing a baffle fitted to the boiler. Is this correct? Im having trouble getting it to fit inside a very tight cupboard with the standard stand off jig fitted. Ive been looking at the sizes on the website only and...
  17. P

    About cheap, compact electric heating taps

    As an alternative to traditional tankless heaters, aliexpress sells a variety of cheap taps incorporating a heater, for example: US $20.2 49% OFF|ATWFS Instant Tankless Water Heater Tap Instantaneous Faucet Kitchen Water Heater Crane Instant Hot Water Faucet Digital EU Plug|hot water...
  18. CorgiDirect

    The compact, easy-to-use testo 340 available from CORGI direct

    The compact, easy-to-use testo 340 available from CORGI direct enables you to carry out emission measurements on industrial combustion and power generation plants both accurately and flexibly VIEW HERE Emission checks in industry: Professional technology meets practical design. The testo...
  19. S

    Help with vokera compact A rf controller

    Hello, everything is working with my rf controller except the comfort temp setting. It's set to 15. When I turn the dial it occasionally jumps to 15.2 then back again. The dial works on all other aspects of selecting things, timer, modes etc. As it's set to a lower temp than what it is, it won't...
  20. Pricklebob

    New compact ensuite: Concealed macerator needed to fit in stud wall

    We are renovating our house and after a lot of playing around I have worked out the best floorplan for upstairs. It means the only place I can fit our ensuite is in the middle of the house, the toilet will be 2.7m from the main soil stack in the family bathroom. It is in a straight line and I...
Creating content since 2001. Untold Media.

OFFICIAL SPONSORS

OFFICIAL SPONSORS

Back
Top
AdBlock Detected

We get it, advertisements are annoying!

Sure, ad-blocking software does a great job at blocking ads, but it also blocks useful features of our website. For the best site experience please disable your AdBlocker.

I've Disabled AdBlock