# Monthly Archives: October 2011

## (Ir)rational Behavior of Calculation

We had the following set of exercises with a group of secondary school students (in Ressun lukio). Assume that a and b are positive real numbers. Then assume one of the following: (i) a and b are both rational (ii) … Continue reading

Posted in Algebra, Calculus, Education, Mathematics, Wednesday Problem | 1 Comment

## The Product of Topological Spaces Does Not Obey Cancellation

Exercise 1. Find metric topological spaces $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, but $$A\times C$$ is homeomorphic to $$B\times C$$. Exercise 2. Find path connected metric topological spaces $$A,B,C$$ such that $$A$$ is not homeomorphic to $$B$$, … Continue reading

Yesterday in a discussion in Komero we concluded that if f is an injective (one-to-one) map from the unit interval [0,1] to the Euclidean plane (or $$\mathbb{R}^n$$), then the interval is homeomorphic with its image. The proof is as follows: … Continue reading