Subspace topology

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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.

Examples

In the following, R represents the real numbers with their usual topology.

  • The subspace topology of the natural numbers, as a subspace of R, is the discrete topology.
  • The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 for example is not an open set in Q). 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 x with a < x < b is both open and closed.
  • The set [0,1] as a subspace of R is both open and closed, whereas as a subset of R it is only closed.
  • As a subspace of R, [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 R. Then [0, 1/2) 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.

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.

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 .

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.

See also

References

  • Bourbaki, Nicolas, Elements of Mathematics: General Topology, Addison-Wesley (1966)
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  • Willard, Stephen. General Topology, Dover Publications (2004) ISBN 0-486-43479-6