families Simple connectivity Theorem. For an open domain the following conditions are equivalent: is simply connected; the integral of every holomorphic function around a closed piecewise smooth curve in vanishes; every holomorphic function in is the derivative of a holomorphic function; every nowhere-vanishing holomorphic function on has a holomorphic logarithm; every nowhere-vanishing holomorphic function on has a holomorphic square root; for any , the winding number of for any piecewise smooth closed curve in is ; the complement of in the extended complex plane is connected. (1) ⇒ (2) because any continuous closed curve, with base point , can be continuously deformed to the constant curve . So the line integral of over the curve is . (2) ⇒ (3) because the integral over any piecewise smooth path from to can be used to define a primitive. (3) ⇒ (4) by integrating along from to to give a branch of the logarithm. (4) ⇒ (5) by taking the square root as where is a holomorphic choice of logarithm. (5) ⇒ (6) because if is a piecewise closed curve and are successive square roots of for outside , then the winding number of about is times the winding number of about . Hence the winding number of about must be divisible by for all , so it must equal . (6) ⇒ (7) for otherwise the extended plane can be written as the disjoint union of two open and closed sets and with and bounded. Let be the shortest Euclidean distance between and and build a square grid on with length with a point of at the centre of a square. Let be the compact set of the union of all squares with distance from . Then and does not meet or : it consists of finitely many horizontal and vertical segments in forming a finite number of closed rectangular paths . Taking to be all the squares covering , then equals the sum of the winding numbers of over , thus giving . On the other hand the sum of the winding numbers of about equals . Hence the winding number of at least one of the about is non-zero. (7) ⇒ (1) This is a purely topological argument. Let be a piecewise smooth closed curve based at . By approximation γ is in the same homotopy class as a rectangular path on the square grid of length based at