Monthly Archives: October 2011

(Ir)rational Behavior of Calculation

Posted on October 18, 2011 by Vadim Kulikov

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

Posted on October 12, 2011 by Vadim Kulikov

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

Posted in Mathematics, Topology, Wednesday Problem | Leave a comment

Wild Embeddings

Posted on October 11, 2011 by Vadim Kulikov

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 [tex]\mathbb{R}^n[/tex]), then the interval is homeomorphic with its image. The proof is as follows: … Continue reading →

Posted in Algebraic Topology, Mathematics, Topology | Leave a comment

Progress

Posted on October 4, 2011 by Vadim Kulikov
Posted in Philosophy, Popular | 3 Comments