# Category Archives: Topology

## 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

## Wild Embeddings

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

## Class-metric

In two weeks I start lecturing Topology I at our University. I am excited. This course was my favourite among the basic undergraduate courses. The course concentrates on metric topology and its goal is to prove simple results about complete … Continue reading

## Map Colouring Problem And Compactness

Suppose G is a planar graph embedded into the plane. The graph divides the plane into regions. Let us say that two regions are adjoint if they have a common edge. Question: (Q) Is it possible to colour the regions … Continue reading

## Ambient Isotopy

Note: One is able to do puzzles 2 and 3 without reading or understanding the text before them. In knot theory, people define equivalence of knots using the concept of ambient isotopy. Two knots (embeddings of the unit circle into … Continue reading

## Around Jordans Curve Theorem I

There are more or less three theorems that are often called Jordan Curve Theorems, while there is a distinction between them. Let us denote by E the Euclidean plane, $$E=\mathbb{R}^2$$ and by $$E^n$$ the Euclidean n-dimensional space, $$E^n=\mathbb{R}^n$$. The Jordan … Continue reading