In

if_and_only_if
In_logic
Logic_is_an_interdisciplinary_field_which_studies_truth_and_reasoning
Reason_is_the_capacity_of__consciously_making_sense_of_things,_applying_logic
Logic_(from_Ancient_Greek,_Greek:__grc,_wikt:λογική,_λογική,_l_...

_the_composite_map_$i\backslash circ\_f$_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_$f:X\backslash to\_Y$_is_continuous_then_the_restriction_to_$S$_is_continuous.
*_If_$f:X\backslash to\_Y$_is_continuous_then_$f:X\backslash to\_f(X)$_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_$S\backslash in\backslash tau$)._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_$X\backslash setminus\_S\backslash in\backslash tau$)._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 $\backslash mathbb$. Then [0, ) is open in ''S'' but not in $\backslash mathbb$. Likewise [, 1) is closed in ''S'' but not in $\backslash mathbb$. ''S'' is both open and closed as a subset of itself but not as a subset of $\backslash mathbb$.

for $X$ then $B\_S\; =\; \backslash $ 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.
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 structure, structu ...

and related areas of 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 general consensus abo ...

, 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
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 structure, structu ...

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 $(X,\; \backslash tau)$ and asubset
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
:$\backslash tau\_S\; =\; \backslash lbrace\; S\; \backslash cap\; U\; \backslash mid\; U\; \backslash in\; \backslash tau\; \backslash rbrace.$
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
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

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 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). I ...

in $(X,\; \backslash tau)$. If $S$ is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of $(X,\; \backslash tau)$. 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 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 ...

:$\backslash iota:\; S\; \backslash hookrightarrow\; X$
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 $\backslash iota$ 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 $\backslash iota$ is continuous. The open sets in this topology are precisely the ones of the form $\backslash iota^(U)$ for $U$ open in $X$. $S$ is then homeomorphic
In the field of , a homeomorphism, topological isomorphism, or bicontinuous function is a between s that has a continuous . Homeomorphisms are the s in the —that is, they are the that preserve all the of a given space. Two spaces with ...

to its image in $X$ (also with the subspace topology) and $\backslash iota$ 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 $\backslash iota$ 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 $\backslash iota$ is a closed 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 ...

.
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 $(X,\; \backslash tau)$ 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 $(S,\backslash tau\_S)$ and $(X,\backslash tau)$ as the topological spaces, related as discussed above. So phrases such as "$S$ an open subspace of $X$" are used to mean that $(S,\backslash tau\_S)$ is an open subspace of $(X,\backslash tau)$, in the sense used below; that is: (i) $S\; \backslash in\; \backslash tau$; and (ii) $S$ is considered to be endowed with the subspace topology.Examples

In the following, $\backslash mathbb$ represents thereal number
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 with their usual topology.
* The subspace topology of the natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...

s, as a subspace of $\backslash mathbb$, 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
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 ...

s $\backslash mathbb$ considered as a subspace of $\backslash mathbb$ do not have the discrete topology ( for example is not an open set in $\backslash mathbb$). 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 $\backslash mathbb$ is both open and closed, whereas as a subset of $\backslash mathbb$ it is only closed.
* As a subspace of $\backslash mathbb$, , 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, wh ...

∪ , 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'' = Properties

The subspace topology has the following characteristic property. Let $Y$ be a subspace of $X$ and let $i\; :\; Y\; \backslash to\; X$ be the inclusion map. Then for any topological space $Z$ a map $f\; :\; Z\backslash to\; Y$ is continuousif and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

the composite map $i\backslash circ\; f$ 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 $f:X\backslash to\; Y$ is continuous then the restriction to $S$ is continuous.
* If $f:X\backslash to\; Y$ is continuous then $f:X\backslash to\; f(X)$ 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 $S\backslash in\backslash tau$). 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 $X\backslash setminus\; S\backslash in\backslash tau$). 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 ba ...Preservation of topological properties

If a topological space having sometopological propertyIn 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 structu ...

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 such that every intersection of a countable collection of Open set, open dense sets in the space is also dense.
Complete metric spaces and locally compact Hausdorff spaces are examples of Baire ...

is a Baire space.
* Every closed subspace of a compact space
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 ...

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 neighbourhood (mathematics), neighbourhoods of each which are disjoint sets, disj ...

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.
See also

* the dual notion quotient space *product topology
Product may refer to:
Business
* Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a market
Market may refer to:
*Market (economics)
*Market economy
*Mark ...

* 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