Subspace topology

This article is about a basic definition in topology.
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Definition

Let $(X, τ)$ be a topological space (viz, a set $X$ endowed with a topology $τ$ ) and $A$ be a subset of $X$ . The subspace topology or induced topology or relative topology on $A$ can be defined in many equivalent ways. Note that $A$ induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.

Definition in open set terms

The subspace topology on $A$ is as follows: a subset of $A$ is open in $A$ if and only if it is the intersection with $A$ of an open subset of $X$ . In other words, $U$ is open in $A$ if and only if there exists an open subset $V$ of $X$ such that $U = V \cap A$ .

(Note that $V$ is not uniquely determined by $U$ ).

Definition in closed set terms

The subspace topology on $A$ is as follows: a subset of $A$ is closed in $A$ if and only if it is the intersection with $A$ of a closed subset of $X$ . In other words, $C$ is closed in $A$ if and only if there exists a closed subset $D$ of $X$ such that $C = D \cap A$ .

(Note that $D$ is not uniquely determined by $C$ ).

Definition in basis terms

Given a basis for $τ$ , we can directly use it to define a basis for the subspace topology on $A$ . Namely, for each basis open set, replace it by its intersection with $A$ .

In other words, if ${U_{i}}_{i \in I}$ form a basis for $τ$ , then a basis for the subspace topology on $A$ is given by ${U_{i} \cap A}_{i \in I}$ .

Definition in subbasis terms

Given a subbasis for $τ$ , we can directly use it to define a basis for the subspace topology on $A$ . Namely, for each subbasis open set, replace it by its intersection with $A$ .

In other words, if ${U_{i}}_{i \in I}$ form a subbasis for $τ$ , then a basis for the subspace topology on $A$ is given by ${U_{i} \cap A}_{i \in I}$ .

Definition in terms of maps

The subspace topology is the coarsest topology that can be endowed to $A$ , for which the inclusion map from $A$ to $X$ , is a continuous map.

Equivalence of definitions

Further information: Equivalence of definitions of subspace topology

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 88-89
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 10 (Theorem 6): introduced under the name relative topology