subspace topology

Definition

If $X$ is a topological space and $S \subseteq X$ is an arbitrary subset, define the subspace topology on $S$ via declaring subset $U \subseteq S$ to be open in $S$ if and only if there exists an open subset $V \subseteq X$ such that $U = V \cap S$.

A subset of $S$ that is open or closed in this subset topology said to be relatively open or relatively closed in $S$.

Any subset of $X$ endowed with the subspace topology is said to be a subspace.

References

1. J. M. Lee, Introduction to smooth manifolds, Second edition. in Graduate texts in mathematics, no. 218. New York Heidelberg Dordrecht London: Springer, 2013, p. 601-602.