Let $T = (X, \mathcal{T})$ be a D1106: Topological space.
An D548: Ordered pair $T_E = (E, \mathcal{T}_E)$ is a topological subspace of $T$ if and only if

(1)	$E \subseteq X$ (D78: Subset)
(2)	$\mathcal{T}_E = \{ U \cap E \mid U \in \mathcal{T} \}$ (D2018: Subspace topology)

We then say that $E$ forms a topological subspace of $T$.