Commutativeness of Fundamental Groups of Topological Groups

The notation and terminology used in this paper have been introduced in the following articles: [3], [19], [9], [10], [16], [20], [4], [5], [22], [23], [21], [1], [6], [17], [18], [2], [25], [26], [24], [15], [12], [13], [8], [14], and [7]. Let A be a non empty set, x be an element, and a be an element of A. Let us observe that (A 7−→ x)(a) reduces to x. Let A, B be non empty topological spaces, C be a set, and f be a function from A×B into C. Let b be an element of B. Let us note that the functor f(a, b) yields an element of C. Let G be a multiplicative magma and g be an element of G. We say that g is unital if and only if (Def. 1) g = 1G. One can check that 1G is unital. Let G be a unital multiplicative magma. Let us note that there exists an element of G which is unital. Let g be an element of G and h be a unital element of G. One can check that g · h reduces to g. One can check that h · g reduces to g. Let G be a group. One can verify that (1G) reduces to 1G. The scheme TopFuncEx deals with non empty topological spaces S, T and a non empty set X and a binary functor F yielding an element of X and states that