Let X be a topological space with topology τ , and let A be a subset of X.

Let τ_A be the collection of all subsets of A that are of the form V ∩ A for V ∈ τ . Then τ_A is a topology on the set A. (It is a straightforward exercise to verify that the topological space axioms are satisfied.) The topology τ_A on A is referred to as the subspace topology on A.

Any subset of a Hausdorff space is itself a Hausdorff space (with respect to the subspace topology).

Let X be a metric space with distance function d, and let A be a subset of X. It is not difficult to prove that a subset W of A is open with respect to the subspace topology on A if and only if, given any point w of W, there exists some δ > 0 such that

                                {a ∈ A : d(a, w) < δ} ⊂ W.

Thus the subspace topology on A coincides with the topology on A obtained on regarding A as a metric space (with respect to the distance function d).

Example: Let X be any subset of n-dimensional Euclidean space Rⁿ . Then the subspace topology on X coincides with the topology on X generated by the Euclidean distance function on X. We refer to this topology as the usual topology on X.

Let X be a topological space, and let A be a subset of X. One can readily verify the following:—

• a subset B of A is closed in A (relative to the subspace topology on A)  if and only if B = A ∩ F for some closed subset F of X;

• if A is itself open in X then a subset B of A is open in A if and only if it is open in X;