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what happened then?

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Who cares about topology?   (Old version)

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wow
https://www.britannica.com/science/topology/History-of-topology

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Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré, although many topological ideas had found their way into mathematics during the previous century and a half. The Latin phrase analysis situs may be translated as "analysis of position" and is similar to the phrase geometria situs, meaning "geometry of position," used in 1735 by the Swiss mathematician Leonhard Euler to describe his solution to the Königsberg bridge problem. Euler's work on this problem also is cited as the beginning of graph theory, the study of networks of vertices connected by edges, which shares many ideas with topology.

During the 19th century two distinct movements developed that would ultimately produce the sibling specializations of algebraic topology and general topology. The first was characterized by attempts to understand the topological aspects of surfacelike objects that arise by combining elementary shapes, such as polygons or polyhedra. One early contributor to combinatorial topology, as this subject was eventually called, was the German mathematician Johann Listing, who published Vorstudien zur Topologie (1847; "Introductory Studies in Topology"), which is often cited as the first print occurrence of the term topology. In 1851 the German mathematician Bernhard Riemann considered surfaces related to complex number theory and, hence, utilized combinatorial topology as a tool for analyzing functions. The German geometers August Möbius and Felix Klein published works on "one-sided" surfaces in 1858 and 1882, respectively. Möbius's example, now known as the Möbius strip, may be constructed by gluing together the ends of a long rectangular strip of paper that has been given a half twist. Surfaces containing subsets homeomorphic to the Möbius strip are called nonorientable surfaces and play an important role in the classification of two-dimensional surfaces. Klein provided an example of a one-sided surface that is closed, that is, without any one-dimensional boundaries. This example, now called the Klein bottle, cannot exist in three-dimensional space without intersecting itself and, thus, was of interest to mathematicians who previously had considered surfaces only in three-dimensional space.

Work by many mathematicians, including the four mentioned above, preceded the 1895 publication of Analysis Situs, in which Poincaré established a basic context for using algebraic ideas in combinatorial topology. Combinatorial topology continued to be developed, especially by the German-born American mathematician Max Dehn and the Danish mathematician Poul Heegaard, who jointly presented one of the first classification theorems for two-dimensional surfaces in 1907. Soon thereafter the importance of associating algebraic structures with topological objects was clearly established by, for example, the Dutch mathematician L.E.J. Brouwer and his fixed point theorem. Although the phrase algebraic topology was first used somewhat later in 1936 by the Russian-born American mathematician Solomon Lefschetz, research in this major area of topology was well under way much earlier in the 20th century.

Simultaneous with the early development of combinatorial topology, 19th-century analysts, such as the French mathematician Augustin Cauchy and the German mathematician Karl Weierstrass, investigated Fourier series, in which sequences of functions converged to other functions in a sense similar to convergence of sequences of points in space. From another point of view, mathematicians such as the German Georg Cantor and the French Émile Borel studied the relationship between Fourier series and set theory. Two initiatives arose from these efforts: establishing a rigorous mathematical setting for major problems of analysis and providing a general setting for mathematical ideas related to convergence of sequences. In 1899 the German mathematician David Hilbert proposed an axiomatic setting for general geometry beyond what the ancient Greeks had considered. In 1905 the French mathematician Maurice Fréchet proposed a consistent scheme of axioms for convergence in an abstract set and also axioms for a metric space, which is a set supplied with a distance function (or "metric"). In 1910 Hilbert suggested axioms for neighbourhoods of points in an abstract set, thereby generalizing properties of small disks centred at points in the plane. Finally, the German mathematician Felix Hausdorff in his Grundzüge der Mengenlehre (1914; "Elements of Set Theory") proposed the foundational axiomatic relationships among the metric, limit, and neighbourhood approaches for general spaces ( see Hausdorff space). Although it was not until 1925 that the Russian mathematician Pavel Alexandrov introduced the modern axioms for a topology on an abstract set, the field of general topology was born in Hausdorff's work.

During the period up to the 1960s, research in the field of general topology flourished and settled many important questions. The notion of dimension and its meaning for general topological spaces was satisfactorily addressed with the introduction of an inductive theory of dimension. Compactness, a property that generalizes closed and bounded subsets of n-dimensional Euclidean space, was successfully extended to topological spaces through a definition involving "covers" of a space by collections of open sets, and many problems involving compactness were solved during this period. The metrization problem, which sought a topological description of the spaces for which the topology could be induced by a metric, was settled following considerable work on the notion of paracompactness, a property that generalizes compactness.

Since the 1960s, research in general topology has moved into several new areas that involve intricate mathematical tools, including set theoretic methods. In the late 1960s researchers worked to generalize some of the topological properties of infinite-dimensional Hilbert space. These efforts foreshadowed a new area of topology now referred to as infinite-dimensional topology. Another major area of modern interest is set theoretic topology, in which the connection between topological spaces and notions from set theory and logic is studied. Some of the problems in this area involve topological propositions that are independent of and yet consistent with the usually assumed axioms of set theory ( see the Click Here to see full-size table table). The resulting arguments, referred to as forcing theory, have yielded provisional truth of some major longstanding topological conjectures.
Stephan C. Carlson

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Mathematicians associate the emergence of topology as a distinct field of mathematics with the 1895 publication of Analysis Situs by the Frenchman Henri Poincaré, although many topological ideas had found their way into mathematics during the previous century and a half.

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The birth of topology │ The History of Mathematics with Luc de Brabandère

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Using topology to solve a counting riddle 

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Using topology to solve a counting riddle 

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Using topology to solve a counting riddle | The Borsuk-Ulam theorem and stolen necklaces

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Topology basics—ArcGIS Pro | Documentation

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Geodatabase topology evolved from the relationships, topological elements, and storage mechanisms of coverages and shapefiles.

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great

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CG101: What is Topology and Why is it important?

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At first, topology can seem like an unusually imprecise branch of mathematics. It's the study of squishy play-dough shapes capable of bending, stretching and compressing without limit. But topologists do have some restrictions: They cannot create or destroy holes within shapes. (It's an old joke that topologists can't tell the difference between a coffee mug and a doughnut, since they both have one hole.) While this might seem like a far cry from the rigors of algebra, a powerful idea called homology helps mathematicians connect these two worlds.

The word "hole" has many meanings in everyday speech - bubbles, rubber bands and bowls all have different kinds of holes. Mathematicians are interested in detecting a specific type of hole, which can be described as a closed and hollow space. A one-dimensional hole looks like a rubber band. The squiggly line that forms a rubber band is closed (unlike a loose piece of string) and hollow (unlike the perimeter of a penny).

Extending this logic, a two-dimensional hole looks like a hollow ball. The kinds of holes mathematicians are looking for - closed and hollow - are found in basketballs, but not bowls or bowling balls.

But mathematics traffics in rigor, and while thinking about holes this way may help point our intuition toward rubber bands and basketballs, it isn't precise enough to qualify as a mathematical definition. It doesn't clearly describe holes in higher dimensions, for instance, and you couldn't program a computer to distinguish closed and hollow spaces.

"There's not a good definition of a hole," said Jose Perea of Michigan State University.

So instead, homology infers an object's holes from its boundaries, a more precise mathematical concept. To study the holes in an object, mathematicians only need information about its boundaries.

The boundary of a shape is the collection of the points on its periphery, and a shape's boundary is always one dimension lower than the shape itself. For example, the boundary of a one-dimensional line segment consists of the two points on either end. (Points are considered zero-dimensional.) The boundary of a solid triangle is the hollow triangle, which consists of one-dimensional edges. Similarly, the solid pyramid is bounded by a hollow pyramid.

If you stick two line segments together, the boundary points where they meet disappear. The boundary points are like the edge of a cliff - they are close to falling off the line. But when you connect the lines, the points that were on the edges are now securely in the center. Separately, the two lines had four total boundary points, but when they are stuck together, the resulting shape only has two boundary points.

If you can attach a third edge and close off the structure, creating a hollow triangle, then the boundary points disappear entirely. Each boundary point of the component edges cancels with another, and the hollow triangle is left with no boundary. So whenever a collection of lines forms a loop, the boundaries cancel.

Loops circle back on themselves, enclosing a central region. But the loop only forms a hole if the central region is hollow, as with a rubber band. A circle drawn on a paper forms a loop, but it is not a hole because the center is filled in. Loops that enclose a solid region - the non-hole kind - are the boundary of that two-dimensional region.

Therefore, holes have two important rigorous features. First, a hole has no boundary, because it forms a closed shape. And second, a hole is not the boundary of something else, because the hole itself must be hollow.

This definition can extend to higher dimensions. A two-dimensional solid triangle is bounded by three edges. If you attach several triangles together, some boundary edges disappear. When four triangles are arranged into a pyramid, each of the edges cancels with another one. So the walls of a pyramid have no boundary. If that pyramid is hollow - that is, it is not the boundary of a three-dimensional solid block - then it forms a two-dimensional hole.

To find all the types of holes within a particular topological shape, mathematicians build something called a chain complex, which forms the scaffolding of homology.

Many topological shapes can be built by gluing together pieces of different dimensions. The chain complex is a diagram that gives the assembly instructions for a shape. Individual pieces of the shape are grouped by dimension and then arranged hierarchically: The first level contains all the points, the next level contains all the lines, and so on. (There's also an empty zeroth level, which simply serves as a foundation.) Each level is connected to the one below it by arrows, which indicate how they are glued together. For example, a solid triangle is linked to the three edges that form its boundary.

Mathematicians extract a shape's homology from its chain complex, which provides structured data about the shape's component parts and their boundaries - exactly what you need to describe holes in every dimension. When you use the chain complex, the processes for finding a 10-dimensional hole and a one-dimensional hole are nearly identical (except that one is much harder to visualize than the other).

The definition of homology is rigid enough that a computer can use it to find and count holes, which helps establish the rigor typically required in mathematics. It also allows researchers to use homology for an increasingly popular pursuit: analyzing data.

That's because data can be visualized as points floating in space. These data points can represent the locations of physical objects, such as sensors, or positions in an abstract space, such as a description of food preferences, with nearby points indicating people who have a similar palate.

To form shapes from data, mathematicians draw lines between neighboring points. When three points are close together, they are filled in to form a solid triangle. When larger numbers of points are clustered together, they form more complicated and higher-dimensional shapes. Filling in the data points gives them texture and volume - it creates an image from the dots.

Homology translates this world of vague shapes into the rigorous world of algebra, a branch of mathematics that studies particular numerical structures and symmetries. Mathematicians study the properties of these algebraic structures in a field known as homological algebra. From the algebra they indirectly learn information about the original topological shape of the data. Homology comes in many varieties, all of which connect with algebra.

"Homology is a familiar construction. We have a lot of algebraic things we know about it," said Maggie Miller of the Massachusetts Institute of Technology.

The information provided by homology even accounts for the imprecision of data: If the data shifts just slightly, the numbers of holes should stay the same. And when large amounts of data are processed, the holes can reveal important features. For example, loops in time-varying data can indicate periodicity. Holes in other dimensions can show clusters and voids in the data.

"There's a real impetus to have methods that are robust and that are pulling out qualitative features," said Robert Ghrist of the University of Pennsylvania. "That's what homology gives you."

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How Mathematicians Use Homology to Make Sense of Topology | Quanta Magazine

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Originally devised as a rigorous means of counting holes, homology provides a scaffolding for mathematical ideas, allowing for a new way to analyze the shapes within data.

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Why do we need topology and what are examples of real-life applications?

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I am very new to topology. I want to know why we need topology in mathematics. What is its significance? Without topology which mathematics concept cannot be explained. What are its real-life

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What is a network topology?

A network topology is the physical and logical arrangement of nodes and connections in a network. Nodes usually include devices such as switches, routers and software with switch and router features. Network topologies are often represented as a graph.

Network topologies describe the arrangement of networks and the relative location of traffic flows. Administrators can use network topology diagrams to determine the best placements for each node and the optimal path for traffic flow. With a well-defined and planned-out network topology, an organization can more easily locate faults and fix issues, improving its data transfer efficiency.

Network geometry can be defined as the physical topology and the logical topology. Network topology diagrams are shown with devices depicted as network nodes and the connections between them as lines. The type of network topology differs depending on how the network needs to be arranged.

Why is network topology important?

Network topology plays a major role in how a network functions. Namely, the topology has a direct effect on network functionality. Choosing the right topology can help increase performance, as a properly chosen and maintained network topology increases energy efficiency and data transfer rates.

A well-defined network topology makes it easier for network admins to locate faults, troubleshoot issues and to allocate network resources. Diagrams are an important reference point in helping to diagnose network issues, as they can represent physical and logical layouts.

What are the types of network topologies?

Network topologies are categorized as either a physical network topology or logical network topology. The physical topology of a network is the physical layout of nodes and connections. Connections include the lines in diagrams that connect nodes, such as Ethernet or Digital Subscriber Line wires, fiber optics and microwaves. Logical network topologies define how a network is set up, including which nodes connect and how, as well as the pattern of data transfers.

This image shows different layouts of network topologies.

There are several types of topologies. For example, physical topologies include the following:

Bus network. In the bus network topology, every node is connected in series along a single cable. This arrangement is found today primarily in cable broadband distribution networks.

Star network. In the star network topology, a central device connects to all other nodes through a central hub. Switched local area networks based on Ethernet switches and most wired home and office networks have a physical star topology.

Ring network. In the ring network topology, the nodes are connected in a closed-loop configuration. Some rings pass data in one direction only, while others are capable of transmission in both directions. These bidirectional ring networks are more resilient than bus networks since traffic can reach a node by moving in either direction. Metro networks based on Synchronous Optical Network technology are the primary example of ring networks.

Mesh network. The mesh network topology links nodes with connections so that multiple paths between at least some points of the network are available. A network is considered to be fully meshed if all nodes are directly connected to all other nodes and partially meshed if only some nodes have multiple connections to others. Meshing multiple paths increases resiliency but also increases cost. However, more space is needed for dedicated links.

Tree network. The tree network topology consists of one root node, and all other nodes are connected in a hierarchy. The topology itself is connected in a star configuration. Many larger Ethernet switch networks, including data center networks, are configured as trees.

Hybrid network. The hybrid network topology is any combination of two or more topologies. Hybrid topologies typically provide exceptional flexibility, as they can accommodate a number of setups. For example, different departments in the same organization may opt for personalized network topologies that are more adaptable to their network needs.

A logical topology for a network refers to the relationship between nodes and logical connections -- defining how data should transfer.

A logical connection differs from a physical path when information can take an invisible hop at intermediate points. In optical networks, optical add-drop multiplexers create logical optical paths because the ADM hop is not visible to the endpoint nodes. Networks based on virtual circuits or tunnels have a physical topology based on the real connection medium -- fiber, for example -- and a logical topology based on the circuits and tunnels.

Sometimes, the logical topology refers to the topology as the user sees it. Internet Protocol ( IP) and Ethernet networks are two common examples. They are fully meshed at the connection level since any user can connect with any other user. This is true unless some means of blocking unwanted connections, like a firewall, is introduced. Full connectivity is a property of the network protocols used -- IP and Ethernet -- not of the network topology itself.

As an example, logical bus and logical ring topologies can be used to define data transmission flows. A logical bus topology features nodes that broadcast data to the entire network. Other nodes on the network check to see if the data is meant for them. Logical ring topology only allows one node to transfer data at a time.

How do you diagram a network topology?

Diagrams of the network topology should be made before constructing a network. This way, network admins know what components constitute the network and how they interact.

This process should start with a list of all the devices in the network. This could, for example, include routers, firewalls and servers. The type of network topology should then be chosen. Once the list of devices is put together and a topology has been chosen, the diagram can be sketched out. Devices should be placed in areas that would best make sense, considering data flows. Next, lines are drawn from network devices. These lines are the connections the network nodes make. Stray away from having too many lines cross over each other and strive to make the diagram clear and easy to read. Scalability and future modifications should also be kept in mind when creating the diagram.

After a sketch showing a rough general input is constructed, diagram software can be used to help map everything out. The software used may include a network diagram template to follow. The network nodes should be named, and lines can be color-coded to further help make the diagram more understandable and clear.

What are examples of network topology tools and software?

There are many network topology tools available, including those that can be categorized as configuration and management tools, network performance software and network mapping software.

For instance, network configuration software helps configure networks, while also automating repetitive tasks. These tools are often used to configure complicated network topologies and can auto detect network nodes and highlight apparent vulnerabilities.

Network performance monitoring and troubleshooting tools keep track of and alert users to network-related performance issues and outages. Some of these tools may keep track of performance using a visual display of the network topology. After setting baseline performance settings, users can track, locate and troubleshoot issues.

Network topology mapping software aids in creating network topology diagrams. Examples of mapping tools and features are the following:

Datadog live network mapping

Edraw

Lucidchart

ManageEngine OpManager network mapping

Microsoft Visio

NetTerrain Logical

SolarWinds Network Topology Mapper

Spiceworks network mapping

Learn four network resiliency factors and what steps organizations can take to build redundancy into their networks.

Continue Reading About network topology

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What Is Network Topology? - Definition from SearchNetworking

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Learn what a network topology is, why it's important, the different types of network topologies and how to diagram them.

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The meme that teaches relativity (Spacetime symmetries and the celestial sphere)