Definition Let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over some topological space X. We say that the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic if there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ with the property that p₁ = p₂ ◦ h.  

We can apply Theorem 1.19 in order to derive a criterion for determining whether or not two covering maps over some connected locally path connected topological space are isomorphic.

Theorem 1.21 Let X be a topological space which is both connected and locally path-connected, let X˜₁ and X˜₂ be connected topological spaces, and let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over X. Let w₁ and w₂ be points of X˜₁ and X˜₂ respectively for which p₁(w₁) = p₂(w₂). Then there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ satisfying p₂ ◦ h = p₁ and h(w₁) = w₂ if and only if the subgroups p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) coincide.

Proof Suppose that there exists a homeomorphism h: X˜₁ → X˜₂ from the covering space X˜₁ to the covering space X˜₂ for which p₂ ◦h = p₁ and h(w₁) = w₂. Thenh_#(π₁(X˜₁, w₁)) = π₁(X˜₂, w₂), and therefore

p_1#(π₁(X˜₁, w₁)) = p_2# (h_#(π₁(X˜₁, w₁))­­) = p_2#(π₁(X˜₂, w₂)).

Conversely suppose that p_1#(π₁(X˜₁, w₁)) = p_2#(π₁(X˜₂, w₂)). It follows from Proposition 1.16 that the covering spaces X˜₁ and X˜₂ are both locally path-connected, since X is a locally path-connected topological space. But X˜₁ and X˜₂ are also connected. It follows from Theorem 1.19 that there exist unique continuous maps h: X˜₁ → X˜₂ and k: X˜₂ → X˜₁ for which p₂ ◦ h = p₁, p₁ ◦ k = p₂, h(w₁) = w₂ and k(w₂) = w₁. But then p₁ ◦ k ◦ h = p₁ and  (k ◦ h)(w₁) = w₁. It follows from this that the composition map k ◦ h is the identity map of X˜₁ (since a straightforward application of Theorem 1.19 shows that any continuous map j: X˜₁ → X˜₁ which satisfies p₁ ◦ j = p₁ and j(w₁) = w₁ must be the identity map of X˜₁). Similarly the composition map h◦k is the identity map of X˜₂. Thus h: X˜₁ → X˜₂ is a homeomorphism whose inverse is k. Moreover p₂ ◦ h = p₂. Thus h: X˜₁ → X˜₂ is a homeomorphism with the required properties.

Corollary 1.22 Let X be a topological space which is both connected and locally path-connected, let X˜₁ and X˜₂ be connected topological spaces, and let p₁: X˜₁ → X and p₂: X˜₂ → X be covering maps over X. Let w1 and w₂ be points of X˜₁ and X˜₂ respectively for which p₁(w₁) = p₂(w₂). Then the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic if and only if the subgroups p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) are conjugate.

Proof Suppose that the covering maps p₁: X˜₁ → X and p₂: X˜₂ → X are topologically isomorphic. Let h: X˜₁ → X˜₂ be a homeomorphism for which p₂ ◦ h = p₁. Then p_1#(π₁(X˜₁, w₁)) = p_2#(π₁(X˜₂, h(w₁))).

It follows immediately from Lemma 1.7 that the subgroups p_1#(π₁(X˜₁, w₁))  and p_2#(π₁(X˜₂, w₂)) of π₁(X, p₁(w₁)) are conjugate.

Conversely, suppose that the subgroups

                      p_1#(π₁(X˜₁, w₁)) and p_2#(π₁(X˜₂, w₂))

of π₁(X, p₁(w₁)) are conjugate. The covering space X˜₂ is both locally path connected (Proposition 1.16) and connected, and is therefore path-connected  (Corollary 1.17). It follows from Lemma 4.7 that there exists a point w of X˜₂ for which p₂(w) = p₂(w₂) = p₁(w₁) and

p_2#(π₁(X˜₂, w)) = p_1#(π₁(X˜₁, w₁)).