In topology and related areas of mathematics, a subspace of a topological space is a subset S of X which is equipped with a topology called the subspace topology (or the relative topology, inherited topology, induced topology, or trace topology).

Definition

and a subset, the subspace topology is defined by

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.

as the coarsest topology for which the inclusion map

is continuous.

is an injection is then homeomorphic is called a topological embedding.

is called an open subspace is an open map. Likewise it is called a closed subspace is a closed map.

Terminology

is considered to be endowed with the subspace topology.

Examples

represents the real numbers with their usual topology.

• The subspace topology of the natural numbers, is the discrete topology.

• The rational numbers 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.

• it is only closed.

• , [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. Then [0, ) is open in S (as for example the intersection between (-, ) and S results in [0, )). Likewise [, 1) is closed in S that can intersect with [0, 1) to result in [, 1)). S.

Properties

is continuous if and only if is continuous. .

.

• is continuous.

• is continuous.

• .

• .

• .

• .

• is a basis.

• 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

Notes

References

• Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)

• Willard, Stephen. General Topology, Dover Publications (2004)