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).[citation needed]

Definition edit

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 edit

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 edit

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   because there is no open subset of   whose intersection with   can result in only the singleton {0}). 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, 12) is open in S but not in   (as for example the intersection between (-12, 12) and S results in [0, 12)). Likewise [12, 1) is closed in S but not in   (as there is no open subset of   that can intersect with [0, 1) to result in [12, 1)). S is both open and closed as a subset of itself but not as a subset of  .

Properties edit

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.

Characteristic property of the subspace topology

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 edit

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.

See also edit

References edit

  • Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
  • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
  • Willard, Stephen. General Topology, Dover Publications (2004) ISBN 0-486-43479-6