# Category Archives: Set Theory

## 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

## Uncountable Borel Sets

Here I promised post some tough stuff. Well, I meant some fruits of our recent research by Hyttinen, Friedman and I. I will not go into the subject here, just post a link to the forthcoming paper. There is an … Continue reading

## Prisoner’s problem 3

Please comment your solutions, questions and remarks.. There are infinitely many prisoners in a prison. The prisoners are taken to the yard and everyone is given a hat which is either black or white. No one knows the colour of … Continue reading

## Nasty Finite Combinatorics

Please comment your solutions, questions and remarks.. I was blamed for having too easy Wednesday problems. So beware! The level is coming up! Suppose that K is a set of n elements, $$n\in\mathbb{N}$$. Suppose that $$K_1,\dots,K_n,K_{n+1}$$ are subsets of K. … Continue reading

## Continuum Hypothesis III

Suppose you pick randomly a real number. What is the probability that it equals to 1? The probability is zero. Suppose $$X\subset [0,1]$$ is a countable subset of the unit interval. What is the probability that a randomly picked real … Continue reading

## Continuum Hypothesis II

Differentiability of Space Filling Curves A Peano curve is a surjective (onto) function $$f\colon\mathbb{R}\to\mathbb{R}^2$$. Apparently such an f cannot be smooth. To see this consider the restrictions of this function to closed intervals $$f\restriction [n,n+1]$$. By smoothness and compactness the … Continue reading