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** (or the **relative topology**, or the **induced topology**, or the **trace topology**).

Given a topological space and a subset of , the **subspace topology** on is defined by

That is, a subset of is open in the subspace topology if and only if it is the intersection of with an open set in . If is equipped with the subspace topology then it is a topological space in its own right, and is called a **subspace** of . Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.

Alternatively we can define the subspace topology for a subset of as the coarsest topology for which the inclusion map

is continuous.

More generally, suppose is an injection from a set to a topological space . Then the subspace topology on is defined as the coarsest topology for which is continuous. The open sets in this topology are precisely the ones of the form for open in . is then homeomorphic to its image in (also with the subspace topology) and is called a topological embedding.

A subspace is called an **open subspace** if the injection is an open map, i.e., if the forward image of an open set of is open in . Likewise it is called a **closed subspace** if the injection is a closed map.

The distinction between a set and a topological space is often blurred notationally, for convenience, which can be a source of confusion when one first encounters these definitions. Thus, whenever is a subset of , and is a topological space, then the unadorned symbols "" and "" can often be used to refer both to and considered as two subsets of , and also to and as the topological spaces, related as discussed above. So phrases such as " an open subspace of " are used to mean that is an open subspace of , in the sense used below -- that is that: (i) ; and (ii) is considered to be endowed with the subspace topology.

In the following, represents the real numbers with their usual topology.

- The subspace topology of the natural numbers, as a subspace of , is the discrete topology.
- The rational numbers considered as a subspace of do not have the discrete topology ({0} for example is not an open set in ). If
*a*and*b*are rational, then the intervals (*a*,*b*) and [*a*,*b*] are respectively open and closed, but if*a*and*b*are irrational, then the set of all rational*x*with*a*<*x*<*b*is both open and closed. - The set [0,1] as a subspace of is both open and closed, whereas as a subset of it is only closed.
- As a subspace of , [0, 1] ∪ [2, 3] is composed of two disjoint
*open*subsets (which happen also to be closed), and is therefore a disconnected space. - Let
*S*= [0, 1) be a subspace of the real line . Then [0, ^{1}⁄_{2}) is open in*S*but not in . Likewise [^{1}⁄_{2}, 1) is closed in*S*but not in .*S*is both open and closed as a subset of itself but not as a subset of .

The subspace topology has the following characteristic property. Let be a subspace of and let be the inclusion map. Then for any topological space a map is continuous if and only if the composite map is continuous.

This property is characteristic in the sense that it can be used to define the subspace topology on .

We list some further properties of the subspace topology. In the following let be a subspace of .

- If is continuous the restriction to is continuous.
- If is continuous then is continuous.
- The closed sets in are precisely the intersections of with closed sets in .
- If is a subspace of then is also a subspace of with the same topology. In other words the subspace topology that inherits from is the same as the one it inherits from .
- Suppose is an open subspace of (so ). Then a subset of is open in if and only if it is open in .
- Suppose is a closed subspace of (so ). Then a subset of is closed in if and only if it is closed in .
- If is a basis for then is a basis for .
- The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

If a topological space having some topological property implies its subspaces have that property, then we say the property is **hereditary**. If only closed subspaces must share the property we call it **weakly hereditary**.

- Every open and every closed subspace of a completely metrizable space is completely metrizable.
- Every open subspace of a Baire space is a Baire space.
- Every closed subspace of a compact space is compact.
- Being a Hausdorff space is hereditary.
- Being a normal space is weakly hereditary.
- Total boundedness is hereditary.
- Being totally disconnected is hereditary.
- First countability and second countability are hereditary.

- the dual notion quotient space
- product topology
- direct sum topology

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.

In mathematics, a topological space is called **separable** if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

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, particularly in topology, an **open set** is an abstract concept generalizing the idea of an open interval in the real line. The simplest example is in metric spaces, where open sets can be defined as those sets which contain a ball around each of their points ; however, an open set, in general, can be very abstract: any collection of sets can be called open, as long as the union of an arbitrary number of open sets in the collection is open, the intersection of a finite number of open sets is open, and the space itself is open. These conditions are very loose, and they allow enormous flexibility in the choice of open sets. In the two extremes, every set can be open, or no set can be open but the space itself and the empty set.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

In the mathematical field of topology, the **Alexandroff extension** is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named for the Russian mathematician Pavel Alexandroff. More precisely, let *X* be a topological space. Then the Alexandroff extension of *X* is a certain compact space *X** together with an open embedding *c* : *X* → *X** such that the complement of *X* in *X** consists of a single point, typically denoted ∞. The map *c* is a Hausdorff compactification if and only if *X* is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the **one-point compactification** or **Alexandroff compactification**. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–

In mathematics, a **base** or **basis** for the topology τ of a topological space (*X*, τ) is a family *B* of open subsets of *X* such that every open set is equal to a union of some sub-family of *B*. For example, the set of all open intervals in the real number line is a basis for the Euclidean topology on because every open interval is an open set, and also every open subset of can be written as a union of some family of open intervals.

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 the area of mathematics known as functional analysis, a **reflexive space** is a locally convex topological vector space (TVS) such that the canonical evaluation map from *X* into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space *X* is reflexive if and only if the canonical evaluation map from *X* into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a *non*-reflexive Banach space that is isometrically isomorphic to its bidual.

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 mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear maps from a bornological space into any locally convex spaces is continuous if and only if it a bounded linear operator.

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, an ** LF-space**, also written

In topology and related fields of mathematics, a **sequential space** is a topological space that satisfies a very weak axiom of countability.

In mathematics, a **polyadic space** is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

In mathematics, the **injective tensor product** of two topological vector spaces was introduced by Alexander Grothendieck and was used by him to define nuclear spaces.

The strongest locally convex topological vector space (TVS) topology on , the tensor product of two locally convex TVSs, making the canonical map *separately* continuous is called the **inductive topology** or the **ι-topology**. When *X ⊗ Y* is endowed with this topology then it is denoted by and called the **inductive tensor product** of *X* and *Y*.

In functional analysis and related areas of mathematics, a **metrizable** topological vector spaces (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

In mathematics, a **convergence space**, also called a **generalized convergence**, is a set together with a relation called a *convergence* that satisfies certain properties relating elements of *X* with the family of filters on *X*. Convergence spaces generalize the notions of convergence that are found in point-set topology. Every topological space gives rise to a canonical convergence but there are convergences, known as *non-topological convergences*, that do not arise from any topological space. Examples of non-topological convergences include convergence in measure and convergence almost everywhere.

- Bourbaki, Nicolas,
*Elements of Mathematics: General Topology*, Addison-Wesley (1966) - Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978],
*Counterexamples in Topology*(Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446 - Willard, Stephen.
*General Topology*, Dover Publications (2004) ISBN 0-486-43479-6

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