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In general topology and related areas of mathematics, the **disjoint union** (also called the **direct sum**, **free union**, **free sum**, **topological sum**, or **coproduct**) 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, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.

The name *coproduct* originates from the fact that the disjoint union is the categorical dual of the product space construction.

Let {*X*_{i} : *i* ∈ *I*} be a family of topological spaces indexed by *I*. Let

be the disjoint union of the underlying sets. For each *i* in *I*, let

be the **canonical injection** (defined by ). The **disjoint union topology** on *X* is defined as the finest topology on *X* for which all the canonical injections are continuous (i.e.: it is the final topology on *X* induced by the canonical injections).

Explicitly, the disjoint union topology can be described as follows. A subset *U* of *X* is open in *X* if and only if its preimage is open in *X*_{i} for each *i* ∈ *I*. Yet another formulation is that a subset *V* of *X* is open relative to *X* iff its intersection with *X _{i}* is open relative to

The disjoint union space *X*, together with the canonical injections, can be characterized by the following universal property: If *Y* is a topological space, and *f _{i}* :

This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map *f* : *X* → *Y* is continuous iff *f _{i}* =

In addition to being continuous, the canonical injections φ_{i} : *X*_{i} → *X* are open and closed maps. It follows that the injections are topological embeddings so that each *X*_{i} may be canonically thought of as a subspace of *X*.

If each *X*_{i} is homeomorphic to a fixed space *A*, then the disjoint union *X* is homeomorphic to the product space *A*×*I* where *I* has the discrete topology.

- Every disjoint union of discrete spaces is discrete
*Separation*- Every disjoint union of T
_{0}spaces is T_{0} - Every disjoint union of T
_{1}spaces is T_{1} - Every disjoint union of Hausdorff spaces is Hausdorff

- Every disjoint union of T
*Connectedness*- The disjoint union of two or more nonempty topological spaces is disconnected

- product topology, the dual construction
- subspace topology and its dual quotient topology
- topological union, a generalization to the case where the pieces are not disjoint

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.

In mathematics, any vector space * has a corresponding ***dual vector space** consisting of all linear forms on *, together with the vector space structure of pointwise addition and scalar multiplication by constants.*

In topology and related areas of mathematics, a **product space** is the Cartesian product of a family of topological spaces equipped with a natural topology called the **product topology**. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

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. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In topology and related branches of mathematics, a topological space is called **locally compact** if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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 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.

In topology and related branches of mathematics, a **totally disconnected space** is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets. In every topological space, the singletons are connected; in a totally disconnected space, these are the *only* connected proper subsets.

In category theory, the **coproduct**, or **categorical sum**, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

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 mathematics, a **pointed space** is a topological space with a distinguished point, the **basepoint**. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as *x*_{0}, that remains unchanged during subsequent discussion, and is kept track of during all operations.

In mathematics, an **adjunction 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. One forms the adjunction space *X* ∪_{f}*Y* by taking the disjoint union of *X* and *Y* and identifying *a* with *f*(*a*) for all *a* in *A*. Formally,

In topology and related areas of mathematics, a **topological property** or **topological invariant** is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space *X* possesses that property every space homeomorphic to *X* possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

In mathematics, a **Banach manifold** is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space. Banach manifolds are one possibility of extending manifolds to infinite dimensions.

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 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 there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to 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, including those that are not metrizable or Hausdorff.

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