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In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a subspace of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''S'' of ''X'' which is equipped with a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of
, the subspace topology on
is defined by
:
That is, a subset of
is open in the subspace topology
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of
with an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
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
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iota ...
:
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
.
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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
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
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
, 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
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
.
Terminology
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 above; that is: (i)
; and (ii)
is considered to be endowed with the subspace topology.
Examples
In the following,
represents the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s with their usual topology.
* The subspace topology of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, as a subspace of
, is the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
.
* The
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s
considered as a subspace of
do not have the discrete topology ( 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
,1as a subspace of
is both open and closed, whereas as a subset of
it is only closed.
* As a subspace of
,
, 1∪
, 3is composed of two disjoint ''open'' subsets (which happen also to be closed), and is therefore a
disconnected space
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 th ...
.
* Let ''S'' =
if_and_only_if_
In__logic_and_related_fields_such_as_mathematics_and_philosophy,_"if_and_only_if"_(shortened_as_"iff")_is_a__biconditional__logical_connective_between_statements,_where_either_both_statements_are_true_or_both_are_false.
The_connective_is__bic_...
_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_then_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_(topology).html" ;"title=", 1) be a subspace of the real line
. Then [0, ) is open in ''S'' but not in
. Likewise [, 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
.
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 if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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 then 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 (topology)">basis for
then
is a basis for
.
* The topology induced on a subset of a metric space by restricting the metric (mathematics), metric to this subset coincides with subspace topology for this subset.
Preservation of topological properties
If a topological space having some
topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spa ...
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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
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 where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...
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. T ...
is weakly hereditary.
*
Total boundedness is hereditary.
* Being
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
is hereditary.
*
First countability
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
and
second countability
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
are hereditary.
See also
* the dual notion
quotient space
*
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 natural-seem ...
*
direct sum topology
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 th ...
References
* Bourbaki, Nicolas, ''Elements of Mathematics: General Topology'', Addison-Wesley (1966)
*
* Willard, Stephen. ''General Topology'', Dover Publications (2004) {{ISBN, 0-486-43479-6
Topology
General topology