In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces . Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and such that .

A **pair of spaces** is an ordered pair (*X*, *A*) where *X* is a topological space and *A* a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of *X* by *A*. Pairs of spaces occur centrally in relative homology,^{ [1] } homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in .

Heuristically, one often thinks of a pair as being akin to the quotient space .

There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space to the pair .

A related concept is that of a triple (*X*, *A*, *B*), with *B* ⊂ *A* ⊂ *X*. Triples are used in homotopy theory. Often, for a pointed space with basepoint at *x*_{0}, one writes the triple as (*X*, *A*, *B*, *x*_{0}), where *x*_{0} ∈ *B* ⊂ *A* ⊂ *X*.^{ [1] }

In mathematics, a **chain complex** is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

**Homological algebra** is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, **general topology** is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is **point-set topology**.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

In algebraic topology, a branch of mathematics, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In category theory, a branch of mathematics, a **pushout** is the colimit of a diagram consisting of two morphisms *f* : *Z* → *X* and *g* : *Z* → *Y* with a common domain. The pushout consists of an object *P* along with two morphisms *X* → *P* and *Y* → *P* that complete a commutative square with the two given morphisms *f* and *g*. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .

In mathematics, and algebraic topology in particular, an **Eilenberg–MacLane space** is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

In algebraic topology, a branch of mathematics, the **(singular) homology** of a topological space **relative to** a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.

In general topology and related areas of mathematics, the **final topology** on a set , with respect to a family of functions into , is the finest topology on that makes those functions continuous.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

In mathematics, specifically in category theory, an **exponential object** or **map object** is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an **exponential law**.

In mathematics, specifically in algebraic topology, the **Eilenberg–Steenrod axioms** are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular homology, developed by Samuel Eilenberg and Norman Steenrod.

In algebraic topology, the **pushforward** of a continuous function : between two topological spaces is a homomorphism between the homology groups for .

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

In mathematics, a **Δ-set***S*, often called a **semi-simplicial set**, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set.

In mathematics, **assembly maps** are an important concept in geometric topology. From the homotopy-theoretical viewpoint, an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly maps correspond to 'assemble' local data over a parameter space together to get global data.

This is a glossary of properties and concepts in algebraic topology in mathematics.

- 1 2 Hatcher, Allen (2002).
*Algebraic Topology*. Cambridge University Press. ISBN 0-521-79540-0.

- Patty, C. Wayne (2009),
*Foundations of Topology*(2nd ed.), p. 276.

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