# 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

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## 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

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

## 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

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## 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

Posted in Combinatorics, Geometry, Logic, Mathematics, Topology | Leave a comment

## 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

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## Even More On BFPT

I wouldn’t mind writing this unless I haven’t already posted two different proofs of Brouwer’s Fixed Point Theorem. Namely a friend of mine and a reader of this blog, exposed me to another way of proving it. This proof has … Continue reading

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

Theorem (Brouwer’s Fixed Point Theorem, BFPT). Suppose that $$B\subset \mathbb{R}^n$$ is the closed n-dimensional unit ball and $$f\colon B\to B$$ is a continuous function. Then there exists a point $$x\in B$$ such that $$f(x)=x$$. Theorem (Jordan’s Curve Theorem) Let $$f\colon … Continue reading Posted in Combinatorics, Games, Mathematics, Topology | | 2 Comments ## Brouwer’s Fixed Point Theorem: Many in One Post In this post I will (1) give a simple proof of Brouwer Fixed Point Theorem (2) fulfill the promise given here (3) present the Wednesday Problem in the form fill in the details in the below text Theorem (Brouwer’s Fixed … Continue reading | | 2 Comments ## TopoLogic The Continuum Hypothesis trilogy will continue later. Today I’ll recall some discussion I made a year ago or so in the student seminar. I guess someone would say that it is surprising, how closely related these two branches of mathematics … Continue reading Posted in Logic, Mathematics, Topology | 2 Comments ## Continued fractions and strange homeomorphisms. Usually one thinks of real numbers represented by a sequence (possibly infinite) of numbers between 0 and 9. For example [tex]34,140414\dots$$ One can use various bases like binary, hexadesimal, but the most common is decimal, although with the growing trend … Continue reading

Posted in Calculus, Mathematics, Topology | 8 Comments