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In algebraic topology, a simplicial *k*-**chain**
is a formal linear combination of *k*-simplices.

Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients typically integers.
The set of all *k*-chains forms a group and the sequence of these groups is called a chain complex.

The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a *k*-chain is a (*k*−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.

**Example 1:** The boundary of a path is the formal difference of its endpoints: it is a telescoping sum. To illustrate, if the 1-chain is a path from point to point , where
,
and
are its constituent 1-simplices, then

**Example 2:** The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Chain_(algebraic_topology)

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Below are some of the main areas studied in algebraic topology:

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

In algebraic topology and abstract algebra, **homology** (in part from Greek ὁμός *homos* "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Algebraic_topology

**Algebraic**
** may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.**

**Algebraic** may also refer to:

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Algebraic

In mathematical logic, an **algebraic definition** is one that can be given using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed.

Saying that a definition is algebraic is a stronger condition than saying it is elementary.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Algebraic_definition

In topology and related branches of mathematics, a **topological space** may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of *open sets*, but the most intuitive is that in terms of *neighbourhoods* and so we give this first.
Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Topological_space

* Topology* was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of

On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published.

In 2007 the former editors of *Topology* announced the launch of the *Journal of Topology*, published by Oxford University Press on behalf of the London Mathematical Society at a significantly lower price. Its first issue appeared in January 2008.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Topology_(journal)

The **topology** of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram. It is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.

Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Topology_(electrical_circuits)

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