Subspace topology

This article is about a basic definition in topology.
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Definition

Let $(X,\tau )$ be a topological space (viz, a set $X$ endowed with a topology $\tau$) 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 $\tau$, 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 $\tau$, 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 $\tau$, 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 $\tau$, 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