Relative topology

of a subset $A$ of a topological space $(X,\tau)$

The system of intersections of all possible open subsets of $(X,\tau)$ (i.e. of elements of the topology $\tau$) with $A$. The relative topology is often called the induced topology.

A subset of the topological space $(X,\tau)$ equipped with the relative topology is called a subspace of $(X,\tau)$. A subspace of a $T_i$-space is itself a $T_i$-space, $i=0,1,2,3,31/2$ (cf. Separation axiom). A subspace of a metrizable space is itself metrizable. Any Tikhonov space of weight $\leq\theta$ is homeomorphic to a subspace of a Hausdorff compactum of weight $\leq\theta$ (Tikhonov's theorem).

References

[a1]	J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 50ff