# Subspace 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 natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

## Definition

Given a topological space (X,τ) and a subset S of X, the subspace topology on S is defined by

$\tau_S = \lbrace S \cap U \mid U \in \tau \rbrace.$

That is, a subset of S is open in the subspace topology if and only if it is the intersection of S with an open set in (X,τ). If S is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of (X,τ). 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 S of X as the coarsest topology for which the inclusion map

$\iota: S \hookrightarrow X$

is continuous.

More generally, suppose i is an injection from a set S to a topological space X. Then the subspace topology on S is defined as the coarsest topology for which i is continuous. The open sets in this topology are precisely the ones of the form i − 1(U) for U open in X. S is then homeomorphic to its image in X (also with the subspace topology) and i is called a topological embedding.

## Examples

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

• The subspace topology of the natural numbers, as a subspace of R, is the discrete topology.
• The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 for example is not an open set in Q). 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 x with a < x < b is both open and closed.
• The set [0,1] as a subspace of R is both open and closed, whereas as a subset of R it is only closed.
• As a subspace of R, $[0,1]\cup[2,3]$ 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 R. Then [0,½) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R.

## Properties

The subspace topology has the following characteristic property. Let Y be a subspace of X and let $i : Y \to X$ be the inclusion map. Then for any topological space Z a map $f : Z\to Y$ is continuous if and only if the composite map $i\circ f$ is continuous.

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

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

• If $f:X\to Y$ is continuous the restriction to S is continuous.
• If $f:X\to Y$ is continuous then $f:X\to f(X)$ is continuous.
• The closed sets in S are precisely the intersections of S with closed sets in X.
• If A is a subspace of S then A is also a subspace of X with the same topology. In other words the subspace topology that A inherits from S is the same as the one it inherits from X.
• Suppose S is an open subspace of X. Then a subspace of S is open in S if and only if it is open in X.
• Suppose S is a closed subspace of X. Then a subspace of S is closed in S if and only if it is closed in X.
• If B is a base for X then $B_S = \{U\cap S : U \in B\}$ is a basis for S.
• The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

## Preservation of topological properties

If whenever a topological space has a certain topological property we have that all of its subspaces share the same property, then we say the topological property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.

• Every open and every closed subspace of a topologically complete space is topologically complete.
• 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.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• subspace topology — Math. See relative topology. * * * …   Universalium

• subspace topology — Math. See relative topology …   Useful english dictionary

• Subspace — may refer to:;Mathematics * Euclidean subspace, in linear algebra, a set of vectors in n dimensional Euclidean space that is closed under addition and scalar multiplication. * Linear subspace, in linear algebra, a subset of a vector space that is …   Wikipedia

• Topology — (Greek topos , place, and logos , study ) is the branch of mathematics that studies the properties of a space that are preserved under continuous deformations. Topology grew out of geometry, but unlike geometry, topology is not concerned with… …   Wikipedia

• subspace — /sub spays /, n. 1. a smaller space within a main area that has been divided or subdivided: The jewelry shop occupies a subspace in the hotel s lobby. 2. Math. a. a subset of a given space. b. Also called linear manifold. a subset of a vector… …   Universalium

• Order topology — In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered set, the order …   Wikipedia

• Glossary of topology — 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… …   Wikipedia

• Extension topology — 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. Contents 1 Extension… …   Wikipedia

• Initial topology — In general topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X which makes those functions continuous.The… …   Wikipedia

• Product 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,… …   Wikipedia