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A covering space is a mathematical concept where a topological space is mapped onto another space, known as the base space, in such a way that the base space is 'covered' by the covering space. This concept can seem abstract, so let's break it down further. In simpler terms, imagine you have a large map (the covering space) placed over a smaller map (the base space). Every point in the smaller map corresponds to multiple points on the larger map, but locally, around any small region of the smaller map, there is a one-to-one correspondence.
In the exercise, we are considering a topological group with neighborhoods that can be lifted to equivalence classes. These equivalence classes form the covering space of the topological group. The idea is to ensure continuity and that local sections (small neighborhoods) are homeomorphically mapped, meaning they retain their topological properties. Understanding covering spaces helps in visualizing how complex structures can be understood by looking at simpler, more manageable parts.

A local homomorphism is a function between two groups that preserves their structure, but only in a confined neighborhood rather than the entire group. It essentially means that within a small region, the function behaves like a perfect homomorphism.
In the exercise, the function \(\phi: U \ \rightarrow \ H\) is a local homomorphism because it satisfies \(\phi(a b)=\phi(a) \phi(b)\) for elements within the small neighborhood U. This preservation of group operation locally is crucial for defining how pairs (V, ψ) behave and interact. By understanding local homomorphisms, we can see how local properties can extend to global properties under certain conditions, such as the simply-connected property of G mentioned in the solution.

An equivalence class is a set of elements that are considered equivalent under a given relation. It's a way of grouping elements that share a common property. The concept helps in simplifying complex structures by categorizing similar elements together.
For the exercise, we define an equivalence relation for pairs (V, ψ) where the pairs are considered equivalent if their functions agree on some smaller neighborhood of a point c in G. Essentially, this means that the pairs behave similarly in these overlapping regions. These equivalence classes then help in forming the covering space of G, as they ensure that locally defined properties are consistent across different regions. This consistency is key to creating a coherent covering space.

A simply-connected space is a type of topological space that has no 'holes' and any loop within the space can be continuously contracted to a single point. This property is significant because it ensures uniqueness in the extension of local properties to global properties.
In our exercise, the group G is simply-connected, which means any local homomorphism like \(\phi\) defined in a small neighborhood U can be extended uniquely to a global homomorphism over the entire group G. This is because in a simply-connected space, path-connectedness and the unique lifting property allow for local behaviors to extend without ambiguity. This concept assures that the function \(\phi\) maintains its homomorphic properties globally, providing a complete and consistent mapping from G to H.