Question

In μnkres' proof of Theorem 54.4, he generalizes to show that the fundamental group of the torus is isomorphic to Z×Z. What is the theorem number 54.4 about?

a) Euclidean Geometry

b) Topology

c) Algebraic Structures

d) Number Theory

Answer 1

μnkres' proof of Theorem 54.4 in Topology shows that the fundamental group of the torus is isomorphic to Z×Z, the direct product of the group of integers with itself. Therefore, the correct option is B.

Step-by-step explanation:

The theorem number 54.4 mentioned in the question is about Topology. In this theorem, μnkres proves that the fundamental group of the torus is isomorphic to Z×Z, which means it is isomorphic to the direct product of the group of integers with itself. The concept of isomorphism in group theory relates the structures of two groups, showing that they have the same algebraic properties. In this case, the torus has a fundamental group that is isomorphic to the group of integers squared.