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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 h ...
and related areas of 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 po ...
''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 o ...
''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 h ...
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, \tau) 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 o ...
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 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 bicond ...
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, their ...
of S 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 ...
in (X, \tau). 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, \tau). 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 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 \iot ...
:\iota: S \hookrightarrow X 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 \iota 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 \iota is continuous. The open sets in this topology are precisely the ones of the form \iota^(U) for U open in X. S 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 isomorph ...
to its image in X (also with the subspace topology) and \iota is called a
topological embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
. A subspace S is called an open subspace if the injection \iota 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 S is open in X. Likewise it is called a closed subspace if the injection \iota 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, a ...
.


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 S is a subset of X, and (X, \tau) is a topological space, then the unadorned symbols "S" and "X" can often be used to refer both to S and X considered as two subsets of X, and also to (S,\tau_S) and (X,\tau) as the topological spaces, related as discussed above. So phrases such as "S an open subspace of X" are used to mean that (S,\tau_S) is an open subspace of (X,\tau), in the sense used above; that is: (i) S \in \tau; and (ii) S is considered to be endowed with the subspace topology.


Examples

In the following, \mathbb 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 r ...
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 \mathbb, 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 to ...
. * 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 ratio ...
s \mathbb considered as a subspace of \mathbb do not have the discrete topology ( for example is not an open set in \mathbb). 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 \mathbb is both open and closed, whereas as a subset of \mathbb it is only closed. * As a subspace of \mathbb,
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
, 3 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
is 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 ...
. * 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_bicond_...
_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_then_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_(so_S\in\tau)._Then_a_subset_of_S_is_open_in_S_if_and_only_if_it_is_open_in_X. *_Suppose_S_is_a_closed_subspace_of_X_(so_X\setminus_S\in\tau)._Then_a_subset_of_S_is_closed_in_S_if_and_only_if_it_is_closed_in_X. *_If_B_is_a_basis_(topology).html" "title=", 1) be a subspace of the real line \mathbb. Then [0, ) is open in ''S'' but not in \mathbb. Likewise [, 1) is closed in ''S'' but not in \mathbb. ''S'' is both open and closed as a subset of itself but not as a subset of \mathbb.


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 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 bicond ...
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 then 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 (so S\in\tau). Then a subset of S is open in S if and only if it is open in X. * Suppose S is a closed subspace of X (so X\setminus S\in\tau). Then a subset of S is closed in S if and only if it is closed in X. * If B is a basis (topology)">basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
for X then B_S = \ is a basis for S. * 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 space ...
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 In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
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", ...
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 many ...
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 In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size� ...
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 and second countability 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-seemi ...
* direct sum topology


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