# Topology

## While geometry is primarily concerned with the mathematical properties of spatial objects, topology is concerned with the mathematical properties of those objects under continuous deformations. Please post all questions about topological subjects like homeomorphisms, manifolds, convergence, and connectedness, as well as their broad applications in computing, physics, and graph theory, into this category.

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Math and Arithmetic
Topology

# What does it mean for a subspace of a topological space to be ''somewhere dense''?

Somewhere dense is defined to be the following:

Let B, t be a topological space and C Ã¢Å â€š B. C is somewhere dense if (Cl C)o Ã¢â€°  ÃƒËœ, the empty set. That is, if the closure of the interior of C has at least one non-empty set.

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Computer Networking
Topology

# What is wireless topology?

It's a network architecture which employs wireless devices. Compared to "classical" network topology, where the cable is network medium, in wireless networks is the air a network medium. It means, that all data travels through the air.

Wireless topology can be of 2 basic categories:

Ad-hoc = no central device, just a bunch of computers connected together through their wireless network adapters.

Infrastructure = there is a central device called "access point" to which all client computers connect. This access point provides interconnectivity between clients and also sometimes between the wireless and wired network.

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Topology

# Divide a square in half 4 different ways?

put a staight line in the middle

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Math and Arithmetic
Topology

# How do you define the least upper bound of a subset?

Let (B, ≤) be a partially ordered set and let CB. An upper bound for C is an element b Є Bsuch that cb for each c Є C. If m is an upper bound for C, and if m ≤ b for each upper bound b of C, then m is a least upper bound of C. C can only have one least upper bound, and it may not have any at all (depending on B). The least upper bound of a set C is often written as lub C.

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Computer Networking
Local Area Network
Topology

# How does a LAN work?

LANs
• A Local Area Network is a small network which is usually contained within one building or campus. It is usually a private network, unlike the public internet. An Administrator in charge controlls file sharing, access and many other factors. LANs can be connected to public networks like the Internet, with some precautions (against hackers, viruses etc). Usually a firewall/proxy server/router acts as the gateway between the LAN and the Public Network. A popular wired LAN technology is the Ethernet (Sometimes called IEEE 802.3). These days Wireless LANs are becoming popular. They are collectively known as IEEE 802.11 LANs.
• I could spend hours going into detail about this question, but I won't. Basically the proxy server will go to internet to pull a webpage for the client requesting it. It also will store a copy of this page (cache) for future requests. Another function of the proxy is that is hides the clients IP address from the "outside world", and uses its own. Therefore, the webmaster of the webpage cant see the IP of the client requesting the page because in actuallity the proxy is requesting the page.

LAN

lan stands for the local area network .ie if you are connecting you comouter or devices (printer ) in a specific limited area ie locally .

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A LAN allows certain computers on the network to offer their resources (hard disks, floppy disks, CD-ROMs, Printers, Modems, etc.) for use by other computers on the network as if they were their own. Computers that offer resources are called Servers.

Computers called Workstations can attach the resources (typically hard disks and printers) offered by servers as if they were their own. For instance, at AA Company, computer #1 has a C: hard disk and a D: CD-ROM. Computer #2 has a C: hard disk and a D: CD-ROM, but computer #2 also attaches computer #1's C: drive as it's own F: drive. To the user of computer #2 it looks as if drive F: is in his own computer. He can use files and programs from the F: drive just as he can from his own C: drive. The network software module that performs this slight of hand is called the redirector.

A computer can be both a Server and Workstations at the same time, in which case it is called a Peer. Networks without dedicated servers are called peer-to-peer networks. Networks with one or more dedicated servers are called server based networks even though they may also have peers on them.

Back to our example. The network computers #1 and #2 are on has a server, computer #3. Computers #1 and #2 each have a copy of an accounting program on them, but both read and write accounting data to their G: drive, which is actually C: on computer #3. The tape backup unit is on computer #3 and backs up all the accounting data for all the computers every night by backing up its own C: drive.

When computers #1 and #2 are using the accounting software that software is running in their own memories. The server is not involved at all except to offer its hard disk for data storage. This server is called a file server.

Since this is a Windows accounting package it is big and slow and swaps to disk a lot, so each computer has it installed on its own hard disk to get decent performance. In the days of small fast DOS programs, workstations would also load the program from the server, so it only had to be installed once in one place.

When computers #1 and #2 do sorted reports, every record has to be read from the server and sorted in the memory of the workstation and written back to temporary files on the server. This causes a lot of network traffic on a larger network.

Lets say AA Company grows a lot and now still has computers #1, #2 and server #3 but has added additional workstations #4 through #29 - and lots of users of the accounting software. all that network traffic causes the network to get really bogged down and users start to complain.

What AA Company does now is ditch that Windows accounting package and install a new multiprocesor Compaq server running Windows NT. The new accounting package uses the Oracle database program to store its data at the server. This new package actually runs on the server (which is now called an application server because it has applications programs running on it). The workstations just have a client program that asks for records and has input and viewing screens. If a client asks for a sorted report all the work is done at the server, cutting network traffic way down. This is called a Client Server network.

Meanwhile, across town, BX Company started with its accounting on a Xenix host computer with some "green screen" terminals wired to it (instead of PCs like AA Company used). There was no network at all, just a lot of serial cables connecting dumb terminals and printers to the host computer.

As it grew, BX upgraded to a Unix host computer and added some PCs that were also wired back to the host and ran terminal emulation software so they could act as terminals to use the accounting. Some of the PCs also got their own printers, which also act as slave printers to the Unix box. Still no network.

Finally, BX Company needed to exchange marketing and project files among the PCs, so they installed a peer-to-peer network connecting all the PCs, and included their big honk'n Sun Enterprise Unix box in the network too. The PCs dumped the terminal emulation package and use telnet which allows them to act as terminals over the network - no more serial cables. Later they add a Linux box to the network to act as a file server, as an Intranet Web server and as a firewall for their DSL connection to the Internet.

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LAN is a Huge topic and to know how it works you need to understand what it features are and in what ways a LAN can work like it can work in a Ring Network,Bus Network...It can be used With WAN from a larger to cover a small area,The transmission can be provided by the Token Passing Technique or the Carrier Sense Multiple Access/Collision Detection Technique...The OSI Layers...The hardwire and the software Required...and so On.

I think you can start from the basics to have a better understanding and don't miss out anything.Search in Google or better buy a Book on Networking.

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Computer Networking
Computer Programming
Local Area Network
Topology

# The consequences if a connection fails Five devices arranged in a mesh topology?

If five devices arranged in a mesh topology so we will have 10 links and 4 I/O ports in each hardware device. If any link goes down from them so it will be easy to find out which one is down and it won't effect on other links. But a bulk of wires and can create problem in re-installation and re-configuration.

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Computer Networking
Local Area Network
Topology

# Cat 3 bus or star topology?

UTP cable would not usually be used in a bus topology. It is very common (even at that low speed) for a star.

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Playstation Network
Topology
Abstract Algebra

# What does the term abelian mean?

The term abelian is most commonly encountered in group theory, where it refers to a specific type of group known as an abelian group. An abelian group, simply put, is a commutative group, meaning that when the group operation is applied to two elements of the group, the order of the elements doesn't matter.

For example:

Let G be a group with multiplication * or addition +. If, for any two elements a, b Ð„ G, a*b = b*a or a + b = b + a, then we call the group abelian.

There are other uses of the term abelian in other fields of math, and most of the time, the idea of commutativity is involved.

The term is named after the mathematician, Niels Abel.

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Computer Networking
Local Area Network
Topology

Hula Hoop!

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Topology

# How do you calculate the horizontal distance in surveying?

2+2

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Math and Arithmetic
Area
Topology

# How many square miles in a 50 mile radius?

If you have a circular area with a 50-mile radius, the area of the circle is approximately 7,854 square miles (rounded).

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Computer Networking
Topology

# Why has the ring topology become obsolete?

There may be various reasons that the ring topology becomes obsolete. One of the reason may be the security . As the information is passed in a ring manner to reach the destination.

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Math and Arithmetic
Proofs
Topology

# How do you prove the Baire Category Theorem?

The Baire Category Theorem is, in my opinion, one of the most incredible, influential, and important results from any field of mathematics, let alone topology. It is known as an existence theorem because it provides the necessary conditions to prove that certain things must exist, even if there aren't any examples of them that can be shown. The theorem was proved by René-Louis Baire in 1899 and is a necessary result to prove, amongst other things, the uniform boundedness principle and the open mapping theorem (two of the three most fundamental results from functional analysis), the real numbers being uncountable, and the existence of continuous, yet nowhere differentiable, functions from R to R.

The proof is quite long and involves some pretty advanced math, so to help with the reader's comprehension there is a list of symbols and their meanings at the end of this proof. Also, I've added many related links with definitions and explanations of the terms used in this proof.

The Baire Category Theorem:

If B, D is a nonempty, complete metric space, then the following two statements hold:

1) If B is formulated as the union of countably-many subsets, C1, C2, …, Cp, then at least one of the Cp is somewhere dense.

2) If A1, A2, …, Ap are countably-many, dense, open subsets of B, then ∩pAp is dense in B, i.e. Cl(∩pAp) = B

Proof:

1) If the first statement is false, then there is a countable family {Cp}, p Є P, of subsets of B such that B = ∪pCp, but (Cl Cp)o = Ø for each p Є P. Therefore, for each p, Cl CpB. Select b1 Є B - Cl C1. There is a positive number m1 < 1, since B - Cl C1 is open, such that N(b1, m1) ⊂ B - Cl C1. Now we set G1 = N(b1, m1/2). Then Cl G1N(b1, m1); hence Cl G1 ∩ Cl C1 = Ø.

Since G1 is a nonempty, open subset of B, that means G1 ⊄ Cl C2. So, choose a b2 Є G1- Cl C2. Since G1 - Cl C2 is open, there is an m2 > 0 such that N(b2, m2) ⊂ G1 - Cl C2. This time we'll require m2 < 1/2 and then set G2 = N(b2, m2/2). Then G2G1 and Cl G2 ∩ Cl C2 = Ø.

If we continue on like this, requiring m3 < 1/3, m4 < 1/4, etc., we'll obtain a decreasing sequence of mp-neighborhoods, G1G2G3 ⊃ … ⊃ Gp ⊃ … such that Cl Gp ∩ Cl Cp = Ø and mp < 1/p. Then Cl G1 ⊃ Cl G2 ⊃ Cl G3 ⊃ … ⊃ Cl Gp ⊃ … and d(Gp) --> 0.

I'm going to use a result from another theorem in topology, not proven here, which says that if G1G2G3 ⊃ … ⊃ Gp ⊃ … , d(Gp) --> 0, and ∩pGp ≠ Ø, then the metric space B, D is complete. Therefore, ∩p Cl Gp ≠ Ø. So, if we pick a g Є ∩pGp, then g Є Cp for some p, since ∪pCp = B. However, that would imply that g Є Cl Cp ∩ Cl Gp which is impossible because Cl Cp and Cl Gp are disjoint. So, for 1), Q.E.D.

2) To start, we're going to suppose that {Ap}, p Є P, is a countable family of dense, open subsets of B. To prove that ∩pAp is dense, all that we need to prove is that every neighborhood of any element of B meets ∩pAp. In other words, for any selected g Є Band any m > 0, we'll show that N(g, m) ∩ (∩pAp) ≠ Ø.

If we set T = Cl N(g, m/2), then TN(g, m). Now we'll show that T ∩ (∩pAp) ≠ Ø. We know that T is a subspace of the closed metric space, B, D, and that Titself is closed. So, using an earlier theorem that won't be proved here, T is a complete metric space. If we set Gp = T - Ap which is equal to Tp ∩ (B - Ap), we see that the intersection of two closed subsets of B, Gp is closed in both B and T.

Now suppose Gp is somewhere dense. Then there is an element t Є T and a number q > 0 such that N(t, q) ∩ T ⊂ Cl GpT = Gp. Therefore, N(t, q) ∩ (T - Gp) = Ø. We can see that t Є T = Cl N(g, m/2). Therefore N(t, q) meets N(g, m/2) at some point z. We then choose q' > 0 such that N(z, q') ⊂ N(t, q) ∩ N(g, m/2). However, since Ap is dense, N(z, q') intersects Ap at a point we'll call z' . Well, then it must be that z' Є N(t, q) ∩ TGp. But, Gp= T - Ap, hence z' Є T - Ap. That implies then that z'Ap which is a contradiction. Therefore Gp must be nowhere dense in T.

So, by the first statement of the theorem, 1), which we already proved, T ≠ ∪pGp. Thus, there is n element s Є T - ∪pGp. Therefore, since Gp = T - Ap, then s Є T ∩ (∩pAp) and so T∩ (∩pAp) ≠ Ø meaning N(g, m) ∩ (∩pAp) ≠ Ø.

Q.E.D.

List of symbols:

R - The set of real numbers, including rational, irrational, positive, and negative numbers, as well as 0. Not including complex numbers having an imaginary part other than 0i, where i is the imaginary number √(-1).

B, D - The metric space of set B with metric D.

pAp - The intersection of all of the subsets A1, A2, …, Ap.

Cl - The closure of whatever set is written after it.

p Є P - p is an element of the set P.

P - The set of all positive integers, not including 0. This set is often referred to as the set of natural numbers and is labeled N, but since at times the natural numbers are said to include 0, I've labeled this set P to avoid ambiguity. Not to mention, I've used N within my label for neighborhood.

pCp - The union of all of the of the subsets C1, C2, …, Cp.

( )o - The interior of whatever set is in the parentheses.

Ø - The empty set; i.e. the set with nothing in it.

N(b1, m1) - The neighborhood of point b1 within distance m1.

⊂ - … is a subset of …

⊃ - … is a superset of …

d( ) - The diameter of whatever set is in the parentheses.

--> 0 - The limit of whatever comes before the symbol "-->" goes to 0.

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Topology

# Merits and demerits of bus topology?

NETWORK TOPOLOGY

Networks can be classified by their topology, which is the basic geometric arrangement of the network. Different types of network configurations exist for network designers to choose from. Communications channels can be connected in different arrangements using several different topologies. This arrangement allows users to exchange information and share resources (software and hardware).

Four basic types of network configurations are star, bus, ring, hierarchical and mesh. Ring, bus, and star topologies are commonly used in LANs and BNs. Star and mesh topologies are commonly used in MANs and WANS. The networks are usually built using a combination of several different topologies.

• Star
• Bus
• Ring
• Hierarchical
• Mesh

Star Topology

A star topology is one in which a central unit provides a link through which a group of smaller computers and devices is connected. The central computer is commonly called a host computer. A host computer is usually a large computer such as a minicomputer or a mainframe. A file server is a large storage device that provides volumes of data and programs to the other units in the network.

In the star network, all interactions between different computers in the network travel through the host computer. The central unit will poll each to decide whether a unit has a message to send. If so, the central computer will carry the message to the receiving computer.

Star networks represent a very popular form of configuration for time-sharing systems in which a central computer makes available resources and databases for several "client" computers to share. As such, the star network is appropriate for systems that demand centralized control. The disadvantage of the star network is that a processing problem in the central computer can be paralyzing to the entire system.

In a star network, the central unit may be a host computer or a file server. The host computer is a large centralized computer, usually a minicomputer or a mainframe. In contrast, the file server is a large-capacity hard-disk storage device. It stores data and programs files shared by the users on the network. Also, called a network server.

Bus Topology

In a bus configuration, each computer in the network is responsible for carrying out its own communications without the aid of a central unit. A common communications cable (the bus) connects all of the computers in the network. As data travels along the path of the cable, each unit performs a query to determine if it is the intended recipient of the message. The bus network is less expensive than the star configuration and is thus widely in use for systems that connect only a few microcomputers and systems that do not emphasize the sharing of common resources.

The problem in a computer on a bus topology does not frustrate the operation of the network, but a crack in the central cable will stop the whole network. Bus topology is popular because many computers can be connected to a single central cable. In a bus topology, each end user computer in the network handles its own communications control. There is no host computer or file server. As the information passes along the bus, it is examined by each terminal to see if the data is for it.

Ring Network

A ring configuration features a network in which each computer is connected to the next two other computers in a closed loop. Like the bus network, no single central computer exists in the ring configuration. Messages are simply transferred from one computer to the next until they arrive at their intended destinations. Each computer on the ring topology has a particular address. As the messages pass around the ring, the computers validate the address. If the message is not addressed to it, the node transmits the message to the next computer on the ring.

This type of network is commonly used in systems that connect widely dispersed mainframe computers. A ring network allows organizations to engage in distributed data processing system in which computers can share certain resources with other units while maintaining control over their own processing functions. However, a failure in any of the linked computers can greatly affect the entire network.

The ring arrangement is the least frequently used with microcomputers. However, as stated above, it often is used to link mainframes over wide geographical areas to build distributed data processing system. The loss of a mainframe usually does not restrain the operation of the network, but a cable problem will stop the network altogether.

Hierarchical TopologyA hierarchical network (or a tree network) resembles a star network in that several computers are connected to a central host computer (usually a mainframe). However, these "client" computers also serve as host computers to next level units. Thus, the hierarchical network can theoretically be compared to a standard organizational chart or a large corporation. Typically, the host computer at the top of the hierarchy is a mainframe computer. Lower levels in the hierarchy could consist of minicomputers and microcomputers. It should be noted that a system can sometimes have characteristics of more than one of the above topologies.

This topology is effective in a centralized corporation. For example, different divisions within a corporation may have individual microcomputers connected to divisional minicomputers. The minicomputers in turn may be connected to the corporation's mainframe, which contains data and programs.

Mesh Topology

This is a net-like communications network in which there are at least two pathways to each node. In a mesh topology, computers are connected to each other by point-to-point circuits. In the topology, one or more computers usually become switching centers, interlinking computers with others.

Although a computer or cable is lost, if there are other possible routes through the network, the damage of one or several cables or computers may not have vital impact except the involved computers. However, if there are only few cables in the network, the loss of even one cable or device may damage the network seriously.

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Topology

# What is the best topology for cyber cafe?

The trivial topology, since in the trivial topology everyone would be 'close' to everyone.

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Topology

# What is Cl short for in topology?

Cl is often used as shorthand for closure, e.g. if B, tis a topological space then Cl B is the closure of B. The closure of a topological space B, t is defined as the intersection of all of the closed sets containing B. The closure of C âŠ‚ B where B, D is a metric space is defined as all of the elements of B that have a 0 metric with C, written as

Cl C = {b Ð„ B | D(b, C) = 0}.

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Computer Network Security
Topology

# What is meant by the term securing your perimeter network security?

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Computer Networking
Topology

# What is dual bus topology?

Start with your basic bus topology, where you have a beginning and an end of the network with however many nodes connected in series between. Now, add an identical bus network, except this time start from the other bus network's end and end at this other network's beginning. That's a dual bus topology. This simply provides a single, fail-safe mechanism to the normal bus topology.

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Computer Networking
Topology

# Why star topology is most preferred one?

Star topology is considered as best due to the following reasons

1.There is a central hub in which each individual system is connected.Suppose if one system fails it does not affect other system.This is the major advantage of star topology.

2.Secondly,it requires minimum cabling for the network

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Science
Computers
Computer Networking
Topology

# Where is mesh topology used?

Mesh topology is used in networks.

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Computer Networking
Topology

# What are consequences if connection fails in 5 devices arrange in a star topology?

Any device that fails in a star topology only disconnects the failing device from the network. The rest of the network is not affected.

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Computer Networking
Computer Terminology
Local Area Network
Topology

# If POOF is an organization want to develop the network total number of computer are 7 which topology is best suitable for the network and justify answer?

According to my point of view, Bus topology is best for this organization because its inexpensive and the bus which can be used must be a high speed so it works well.

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Computer Networking
Local Area Network
Topology

# Which network topology is the best?

The best topology is ring topology.

Star is the next best after ringr, and then bus, which is pretty old, but not too expensive.

Full mesh topology is theoretically the best since every device is connected to every other device, thus maximizing speed and security. These, however, are quite expensive to install. The next best would be tree topology, which is basically a connection of stars.

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Topology

# What axioms must be satisfied for a collection of subsets to be called a topology on a set?

There are three axioms that must be satisfied for a collection of subsets, t, of set B to be called a topology on B.

1) Both B and the empty set, Ã˜, must be members of t.

2) The intersection of any two members of t must also be a member of t.

3) The union of any family of members of t must also be a member of t.

If these axioms are met, the members of t are known as t-open or simply open, subsets of B.

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Computers
Education
Numerical Analysis and Simulation
Topology