TheInfoList

In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... and related areas of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a subspace of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... ''S'' of ''X'' which is equipped with a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... 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 $\left(X, \tau\right)$ and a
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... $S$ 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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it is the
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of $S$ with an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in $\left(X, \tau\right)$. If $S$ is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of $\left(X, \tau\right)$. 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 topologyIn topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies. Definition A topology on a set may be defined as the c ...
for which the
inclusion map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
:$\iota: S \hookrightarrow X$ 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 ga ...
. More generally, suppose $\iota$ is an
injection Injection or injected may refer to: Science and technology * Injection (medicine) An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ... 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 $\iota^\left(U\right)$ for $U$ open in $X$. $S$ is then
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to its image in $X$ (also with the subspace topology) and $\iota$ is called a
topological embedding In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A subspace $S$ is called an open subspace if the injection $\iota$ is an
open map In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
.

# 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 $\left(X, \tau\right)$ 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 $\left(S,\tau_S\right)$ and $\left(X,\tau\right)$ as the topological spaces, related as discussed above. So phrases such as "$S$ an open subspace of $X$" are used to mean that $\left(S,\tau_S\right)$ is an open subspace of $\left(X,\tau\right)$, in the sense used below; 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, as a subspace of $\mathbb$, is the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
. * The
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
, 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 (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conne ...
. * Let ''S'' =
if_and_only_if_ In_logic_ Logic_is_an_interdisciplinary_field_which_studies_truth_and_reasoning._Informal_logic_seeks_to_characterize_Validity_(logic),_valid_arguments_informally,_for_instance_by_listing_varieties_of_fallacies._Formal_logic_represents_st_...
_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_$f:X\to_f\left(X\right)$_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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
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 $f:X\to f\left(X\right)$ 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 basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
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 Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
is a Baire space. * Every closed subspace of a
compact space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is compact. * Being a
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ... is hereditary. * Being a
normal space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical stru ... is weakly hereditary. * Total boundedness is hereditary. * Being
totally disconnectedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
is hereditary. * First countability and second countability are hereditary.