# Monthly Archives: January 2013

## Topology: Continuous Maps

Continuity in Metric Spaces Following the case of real analysis, let’s define continuous functions via the usual ε-δ definition. Definition. Let (X, d) and (Y, d’) be two metric spaces. A function f : X → Y is said to be … Continue reading

## Topology: Disjoint Unions

Disjoint Unions Let X and Y be topological spaces and be a set-theoretic disjoint union. We wish to define a topology on Z in a most natural way. Definition. The topology on is defined to be: It’s almost trivial to check that this … Continue reading

## Topology: Product Spaces (I)

In this article, we consider the product of two topological spaces. To motivate our definition, we first begin with metric spaces (X, dX) and (Y, dY). Letting Z = X × Y be the set-theoretic product, we wish to define a metric on Z from dX and … Continue reading

## Topology: Subspaces

First, suppose (X, d) is a metric space. If Y is a subset of X, then one can restrict the metric to , i.e. for any , we set d’(y, y’) := d(y, y’). It’s not hard to show that the resulting function is a metric on Y. … Continue reading

## Topology: Bases and Subbases

Bases Recall that though a subring or ideal of a ring may be rather huge, it often suffices to specify just a few elements which will generate the subring or ideal. Likewise, in a topology,  one can specify a few … Continue reading

## Topology: Basic Definitions

Motivation and Definition While studying analysis, one notices that many important concepts can be defined in terms of “open sets”. One gets the inkling that this concept is critical in forming our notions of continuity, limits etc. In this article, we … Continue reading

## Burnside’s Lemma and Polya Enumeration Theorem (2)

[ Acknowledgement: all the tedious algebraic expansions in this article were performed by wolframalpha. ] Counting Graphs One of the most surprising applications of Burnside’s lemma and Polya enumeration theorem is in counting the number of graphs up to isomorphism. … Continue reading

## Burnside’s Lemma and Polya Enumeration Theorem (1)

[ Note: this article assumes you know some rudimentary theory of group actions. ] Let’s consider the following combinatorial problem. Problem. ABC is a given equilateral triangle. We wish to colour each of the three vertices A, B and C by … Continue reading

## Basic Analysis: Closed Subsets and Uniform Continuity

Let’s consider another question: suppose f : D → R is continuous, where D is a subset of R. If (xn) is a sequence in D converging to some real L, is it true that (f(xn)) is also convergent? Now if L is in D, then we know that (f(xn)) → (f(L)). … Continue reading