First cohomology group with integer coefficients

Definition
Let $$X$$ be a path-connected space). The first cohomology group with integer coefficients of $$X$$, denoted $$H^1(X)$$ or $$H^1(X;\mathbb{Z})$$ is defined in the following equivalent ways:


 * It is the set of homotopy classes of based maps from $$X$$ to $$S^1$$
 * It is the set of homotopy classes of unbased maps from $$X$$ to $$S^1$$
 * It is the set of fiber bundles over $$X$$ with fiber equal to $$\mathbb{Z}$$ (in other words, it is the covering spaces (possible disconnected) whose fibers are the group $$\mathbb{Z}$$)