and related areas of mathematics
, a subspace
of a topological space X
is a subset S
which is equipped with a natural topology induced from that of X
called the subspace topology
(or the relative topology
, or the induced topology
, or the trace topology
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
with an open set
is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace
. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.
If is open, closed or dense in we call an open subspace, closed subspace or dense subspace of , respectively.
Alternatively we can define the subspace topology for a subset of as the coarsest topology for which the inclusion map
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.
- Given the real numbers with the usual topology the subspace topology of the natural numbers, as a subspace of the real numbers, is the discrete topology.
- The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 is not open in Q).
- Let S = [0,1) be a subspace the real line R. Then [0,½) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R.
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 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 . Then a subspace of is open in if and only if it is open in .
- Suppose is a closed subspace of . Then a subspace of is closed in if and only if it is closed in .
- If is a base 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 whenever a topological space has a certain topological property we have that all of its subspaces share the same property, then we say the topological property is hereditary. If only closed subspaces must share the property we call it weakly hereditary.
- Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
- Wilard, Stephen. General Topology, Dover Publications (2004) ISBN 0-486-43479-6