In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a 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 subset  of , the subspace topology on  is defined by

That is, a subset of  is open in the subspace topology if and only if it is the intersection of  with an open set 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

is continuous.

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 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, 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.

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 numbers with their usual topology.
 The subspace topology of the natural numbers, as a subspace of , is the discrete topology.
 The rational numbers  considered as a subspace of  do not have the discrete topology ({0} 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 [0,1] as a subspace of  is both open and closed, whereas as a subset of  it is only closed.
 As a subspace of , [0, 1] ∪ [2, 3] 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 . 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 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 for  then  is a basis for .
 The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.

Preservation of topological properties 

If a topological space having some topological property 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 is compact.
 Being a Hausdorff space is hereditary.
 Being a normal space is weakly hereditary.
 Total boundedness is hereditary.
 Being totally disconnected is hereditary.
 First countability and second countability are hereditary.

See also
 the dual notion quotient space 
 product topology
 direct sum topology

References 
 Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
 
 Willard, Stephen. General Topology, Dover Publications (2004) 

Topology
General topology