In mathematics, an **adjunction space** (or **attaching space**) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let *X* and *Y* be topological spaces, and let *A* be a subspace of *Y*. Let *f* : *A* → *X* be a continuous map (called the **attaching map**). One forms the adjunction space *X* ∪_{f}*Y* (sometimes also written as *X* +_{f}*Y*) by taking the disjoint union of *X* and *Y* and identifying *a* with *f*(*a*) for all *a* in *A*. Formally,

where the equivalence relation ~ is generated by *a* ~ *f*(*a*) for all *a* in *A*, and the quotient is given the quotient topology. As a set, *X* ∪_{f}*Y* consists of the disjoint union of *X* and (*Y* − *A*). The topology, however, is specified by the quotient construction.

Intuitively, one may think of *Y* as being glued onto *X* via the map *f*.

- A common example of an adjunction space is given when
*Y*is a closed*n*-ball (or*cell*) and*A*is the boundary of the ball, the (*n*−1)-sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex. - Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from
*X*and*Y*before attaching the boundaries of the removed balls along an attaching map. - If
*A*is a space with one point then the adjunction is the wedge sum of*X*and*Y*. - If
*X*is a space with one point then the adjunction is the quotient*Y*/*A*.

The continuous maps *h* : *X* ∪_{f}*Y*→*Z* are in 1-1 correspondence with the pairs of continuous maps *h*_{X} : *X*→*Z* and *h*_{Y} : *Y*→*Z* that satisfy *h*_{X}(*f*(*a*))=*h*_{Y}(*a*) for all *a* in *A*.

In the case where *A* is a closed subspace of *Y* one can show that the map *X* → *X* ∪_{f}*Y* is a closed embedding and (*Y* − *A*) → *X* ∪_{f}*Y* is an open embedding.

The attaching construction is an example of a pushout in the category of topological spaces. That is to say, the adjunction space is universal with respect to the following commutative diagram:

Here *i* is the inclusion map and *ϕ*_{X}, *ϕ*_{Y} are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of *X* and *Y*. One can form a more general pushout by replacing *i* with an arbitrary continuous map *g*—the construction is similar. Conversely, if *f* is also an inclusion the attaching construction is to simply glue *X* and *Y* together along their common subspace.

In topology and related branches of mathematics, a **connected space** is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.

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In mathematics, **weak topology** is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, a **topological vector space** is one of the basic structures investigated in functional analysis.

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In topology and related areas of mathematics, the **quotient space** of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the **quotient topology**, that is, with the finest topology that makes continuous the canonical projection map. In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.

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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 topology and related areas of mathematics, a **subspace** of a topological space *X* is a subset *S* of *X* which is equipped with a topology induced from that of *X* called the **subspace topology**.

In general topology and related areas of mathematics, the **disjoint union** of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the **disjoint union topology**. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.

In mathematics, especially homotopy theory, the **mapping cone** is a construction of topology, analogous to a quotient space. It is also called the **homotopy cofiber,** and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces.

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 mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, an **extension topology** is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.

In topology, a **coherent topology** is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a *topological union* of those subspaces. It is also sometimes called the **weak topology** generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.

In topology, a subject in mathematics, a **graph** is a topological space which arises from a usual graph by replacing vertices by points and each edge by a copy of the unit interval , where is identified with the point associated to and with the point associated to . That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes.

In functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then these points will necessarily converge to some point in the space. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* and *Cauchy filters*, which generalize Cauchy sequences. Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, even those that are not metrizable or Hausdorff. Completeness is an extremely important property for a topological vector space to possess. Prominent examples of complete TVSs that are also metrizable include all Fréchet spaces, Banach spaces, and Hilbert spaces. Prominent examples of complete TVS that are (typically) *not* metrizable include strict LF-spaces and nuclear spaces such as the Schwartz space of smooth functions and also the spaces of distributions and test functions.

In the branch of mathematics called functional analysis, when a topological vector space X admits a direct sum decomposition *X* ≅*Y* ⊕ *Z*, the spaces Y and Z are called *complements* of each other. This happens if and only if the addition map *Y* × *Z* → *X*, which is defined by (*y*, *z*) ↦ *y* + *z*, is a homeomorphism. Note that while this addition map is always continuous, it may fail to be a homeomorphism, which is why this definition is needed.

- Stephen Willard,
*General Topology*, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.*(Provides a very brief introduction.)* - "Adjunction space".
*PlanetMath*. - Ronald Brown, "Topology and Groupoids" pdf available , (2006) available from amazon sites. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes.
- J.H.C. Whitehead "Note on a theorem due to Borsuk" Bull AMS 54 (1948), 1125-1132 is the earliest outside reference I know of using the term "adjuction space".

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