In
topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related areas of
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
subspace of a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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
and a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of
, the
subspace topology on
is defined by
That is, a subset of
is open in the subspace topology
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it is the
intersectionIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of
with an
open setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
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 mapIn mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
is continuous.
More generally, suppose
is an
injectionIn mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
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.
Examples
In the following,
R represents the
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s with their usual topology.
- The subspace topology of the natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, as a subspace of R, is the discrete topology.
- The rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s 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 
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,
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
be a subspace of
and let
be the inclusion map. Then for any topological space
a map
is continuous
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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
.
Preservation of topological properties
If whenever a topological space has a certain
topological propertyIn 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...
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
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
is a Baire space.
- Every closed subspace of a compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
is compact.
- Being a Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
is hereditary.
- Being a normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
is weakly hereditary.
- Total boundedness is hereditary.
- Being totally disconnected is hereditary.
- First countability and second countability are hereditary.