Question

Generalize Exercise 10 to show that the portion of Γ contained in an interval remaining at the nth

stage of the construction of Γ is homeomorphic to Γ.

Answer 1

To generalize Exercise 10 and prove that the portion of Γ contained in an interval remaining at the nth stage of the construction of Γ is homeomorphic to Γ, we can use mathematical induction.

Step-by-step explanation:

In Exercise 10, we are asked to generalize and show that the portion of Γ contained in an interval remaining at the nth stage of the construction of Γ is homeomorphic to Γ.

To prove this, we can use mathematical induction. The base case would be n = 0, where Γ is the original interval. The portion of Γ contained in the interval would also be Γ itself.

For the induction step, assume that the portion of Γ contained in an interval remaining at the k-th stage is homeomorphic to Γ.

Now, let's consider the k+1-th stage. The portion of Γ contained in the interval remaining at this stage would be obtained by removing some points from Γ to form the interval. However, since Γ and the interval from the k-th stage are homeomorphic, we can conclude that the portion of Γ contained in the interval remaining at the k+1-th stage is also homeomorphic to Γ.