what is topology useful for?
150485
Maybe it aids the police:\n\n>>The definition of topology leads to the following mathematical joke (Renteln and Dundes 2005): \n\nQ: What is a topologist? A: Someone who cannot distinguish between a doughnut and a coffee cup.<<
628343
There are a couple of related constructs that are called weak topologies. They are both based on having a set X and several topological spaces Y_a along with functions f_a:X->Y_a. The goal is to make all of thee functions continuous.\n\nBy the definition of continuity, this requires f_a^(-1)[U_a] to be open in X for every U_a open in Y_a. The problem is that the collection of all these inverse images doesn't form a topology. But we can make it a *subbase* for a topology by declaring all finite intersections as basis elements for a topology, so the actual topology consists of all unions of those finite intersections. This is called the weak topology.\n\nIn addition to making all the functions f_a continuous, it turns out that any function g:Z->X is continuous if and only if all f_a &#92;circ g are continuous (the compositions). Also, a net [not just a sequence, like in one link] (x_b) converges in X if and only if each f_a(x_b) converges in Y_a.\n\nThe weak topology construction is used to define infinite product topologies where the functions f_a are the projections to the different factors. It is also an important aspect of functional analysis (like in the second link above), where you have a vector space X and a vector space of linear functionals Y (since each linear functional is a function into either the reals or the complexes, this gives the functions in the weak topology construction). In this case, X becomes a locally convex topological vector space. It also turns out that the continuous linear functionals on X are precisely those in Y.
31705
The best guess today: no.\n\nIn modern cosmology the topology of the universe appears to be open. "Topology" meaning the shape, although in the case of the universe to observe the shape we would have to exist in 4 spatial dimensions. Since we're stuck in 3-dimensions we have to get by on analogies.\n\nThe topology or shape of the universe is something which comes out of Einstein's General Theory of Relativity. It depends primarily on the distribution of mass and energy in the universe. If the amount of mass-energy is more than a certain critical value then the universe would be "closed." Basically the mass would curve space-time back onto itself. In a closed universe you could fly off in the direction of the north pole and (after about 25 billion years) and circle the universe to land on the south pole.\n\nIf the amount of mass in the universe is less than this critical value then the topology of the universe is a bit more complex, but, to answer your question space would be infinite.\n\nBased on what we know today that is the case, space is infinite.\n\nHowever, we cannot see an infinite universe. When we see things that are farther away we also see them from deeped in the past. A galaxy we see that is 10 billion light years distant is actually a view of what the galaxy looked like 10 billion years in the past.\n\nWhen we look farther in space, and thus earlier in the history of the universe, we eventually run into the microwave background. For the first 10,000 years or so after the Big Bang the universe was so dense that it wasn't yet transparent. When the density dropped enough to be transparent then some of the light could escape for us to detect much later.\n\nHowever, while the universe is thought to be about 13 billion years old, we can actually see out a bit farther than that, I think around 20 billion light years, because the universe is expanding.\n\nWhat this boils down to is that there is a sort of "wall" beyond which we can't see. This wall corresponds to a time soon after the Big Bang. However, studies of the total mass-energy within the universe strongly suggest that it has an open topology - that it is infinite.\n\n(Of course it is possible that this infinity will be refined when we have a better theory of how Quantum Mechanics and General Relativity play togther. String Theory is one way to combine these two.)
1128311
Differential geometry/manifolds, Optimization word problems in 3D, topology proofs.
328121
Analytic and Algebraic Number Theory contain easily stated theorems with extensive wide-spanning proofs that require knowledge from almost every area of mathematics, including topology and category theory. As such, I would say this area of study is the most difficult, as one that requires the most mathematical experience to tackle. \nAlgebraic topology, on the other hand, can easily be taught in its basic form to motivated high schoolers. It doesn't even require knowledge of calculus, though such knowledge enriches the subject when one gets to cohomology and applications to vector fields and differential equations. These topics can be saved for a college course.
586831
CORBA is the acronym for Common Object Request Broker Architecture. It is vendor independent Middleware.\n\nSimply put its used to connect different types of systems (Mainframes, Linux, Handhelds, Windows etc..) in one framework.\n\nCORBA is useful in many situations. Because of the easy way that CORBA integrates machines from so many vendors, with sizes ranging from mainframes through minis and desktops to hand-helds and embedded systems, it is the middleware of choice for large (and even not-so-large) enterprises. One of its most important, as well most frequent, uses is in servers that must handle large number of clients, at high hit rates, with high reliability. CORBA works behind the scenes in the computer rooms of many of the world's largest websites; ones that you probably use every day. Specializations for scalability and fault-tolerance support these systems. But it's not used just for large applications; specialized versions of CORBA run real-time systems, and small embedded systems.
873938
I'm assuming A is not empty. Then, by the definition of the topology T, it follows A is in T, so that A is open and intB = B. Right.\n\nclB = B? No! B is a subset of clB and since B is not empty, for it contains non-empty A, it follows clB is not empty. Wrong.\n\nNow, see. In the topology T, a set S is closed if it's complement is open. Therefore, S is closed if  S = X or if the complement of S contains A, which, in this case, implies S doesn't intersect A,. clB is the intersection f all closed sets containing B. So, the only closed set that contains B is X, so that clB = X.\n\nWe have that Fr(B) = clB - intB = X - B, so that Fr(B) is the complement of B. therefore, unless B = X, your guess is wrong.\n\nJust check, I may have made a mistake.
81666
Planar topology requires that all lines start and end in nodes and no two lines cross,as with polygonal objects. However, polygon information is not maintained. Withthe exception of polygon filling, planar and polygonal objects appear the same. Pla-nar topology may be appropriate for hydrology if no lakes are present. Planar topol-ogy may also be appropriate for road systems that lack underpasses and overpassesor other features that require network topology for correct representation. Note thepresence of nodes (red) at every position where lines would otherwise cross, as wellas at the dangling ends, in the roads at the left
683684
What is it supposed to be used for?
608757
internet is useful these days.like geeting information very quickly from one compute to another .i has many benifit like railway reservation,sending mail,watching movie,your bank status checking,travel info ,it has been very good resource .so i can say that it is very useful in our life
1171866
Its better to have VLANs and interconnect them via routers or Layer 3 switches.\n\nIts not good to have isolated islands for each dept or area, since nowadays resources like internet bandwidth have to be shared among multiple users and locations.
476997
The Zariski topology is a topology on R^n (more generally on F^n for some field). The closed sets are defined to be sets of the form {x in R^n: p(x)=0} where p is a polynomial. That this does define a topology follows from properties of polynomials. On R, this is exactly the co-finite topology, but on R^n with n>=2, this is not the case. The main difficulty with this topology is that it is never Hausdorff, so limits need not be unique. It does reflect a lot of the geometry of algebraic varieties, though.
106674
Topology As in network topology. The geometric physical or electrical configuration describing a local communication net-work; the shape or arrangement of a system. The most common topologies are the bus, ring and star.\n\nInsulation Displacement Connection (IDC) A type of wire termination in which the wire is "punched down" into a metal holder which cuts into the insulation wire and makes contact with the conductor, thus causing the electrical connection to be made.\n\nIDF Intermediate Distribution Frame. A metal rack designed to connect cables and located in an equipment room or closet. Consists of components that provide the connection between inter-building cabling and the intra-building cabling, i.e. between the Main Distribution Frame (MDF) and individual phone wiring. There’s usually a permanent, large cable running between the MDF and IDF. The changes in wiring are done at the IDF. This saves confusion in wiring.\n\nHome Run Phone system wiring where the individual cables run from each phone directly back to the central switching equipment. Home run cabling can be thought of as "star" cabling. Every cable radiates out from the central equipment. All PBXs and virtually all key systems work on home run cabling. Some local area networks work on home run wiring. See also Star Wiring, Daisy Chain.\n\nDemarcation Point The point of interconnection between telephone company facilities and your building wiring. The demarcation point ("demarc") shall be located on the subscriber’s side of the telephone company’s protector, or the equivalent thereof in cases where a protector is not required.\n\nCrossconnect Distribution system equipment used to terminate and administer communication circuits. In a wire crossconnect, jumper wires or patch cords are used to make circuit connections. In an optical crossconnect fiber patch cords are used. The crossconnect is located in an equipment room, riser closet, or satellite closet.\n\nConnecting Block Also called a terminal block, a punch-down block, a quick-connect block, a crossconnect block. A plastic block containing metal wiring terminals to establish connections from one group of wires to another. Usually each wire can be connected to several other wires in a bus or common arrangement. There are several types of connecting blocks: 66 clip, BIX, Krone, 110, etc. A connecting block has insulation displacement connections (IDCs), which means you don’t have to remove insulation from around the wire conductor before you "punch it down" (terminate it).
730740
Sure, absolutely.\n\nMATLAB is vector & matrix based, which is useful for many kinds of problems.\n\nThere are many built in functions, for probability and statistics, maybe even for your specific application.  Check mathworks.com\n\nThe graphing capabilities are excellent for visualizing data.\n\nAlso checkout Simulink.
4814
Vampire Clamps where mainly used for older 10base5 ethernet networks. The system was a bus topology which meant the cable was one long run. To connect computers to the bus cable a vampire clamp would be used. It would clamp around the cable with a spike in the middle to puncture the cable into the wire core to connect the PC.
41063
Use cases are part of the Unified Modeling Language.\n\nA use case illustrates a unit of functionality provided by the system. The main purpose of the use-case diagram is to help development teams visualize the functional requirements of a system, including the relationship of "actors" (human beings who will interact with the system) to essential processes, as well as the relationships among different use cases.
19870
For one, OPML is useful for transferring your list of blogs you read from one reader to another. Or you can use it to share with others. \n\nI've seen meeting notes stored as OPML. I've also seen it used to store a list of RSS feeds that are later used for crawling for application.