Topological problems are, in general, too complicated to be solved directly. A rather crude method to make them more manageable is to discretise them by replacing the spaces with the sets of their path components. Finer discretisation methods are obtained by first replacing the spaces with auxiliary spaces and then moving on to the path components. The resulting sets often have an algebraic structure, making their description easier. We will hint at this at the end of this chapter using the example of the circle and then exploit it further in the following chapters.