# Monthly Archives: March 2013

## Topology: More on Algebra and Topology

We’ve arrived at the domain where topology meets algebra. Thus we have to proceed carefully to ensure that the topology of our algebraic constructions are well-behaved. Let’s look at topological groups again. Our first task is to show that the … Continue reading

## Topology: Quotients of Topological Groups

Topology for Coset Space This is really a continuation from the previous article. Let G be a topological group and H a subgroup of G. The collection of left cosets G/H is then given the quotient topology. This quotient space, however, satisfies an additional … Continue reading

## Topology: Quotient Topology and Gluing

In topology, there’s the concept of gluing points or subspaces together. For example, take the closed interval X = [0, 1] and glue the endpoints 0 and 1 together. Pictorially, we get: That looks like a circle, but to prove it’s … Continue reading

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## Topology: Topological Groups

This article assumes you know some basic group theory. The motivation here is to consider groups whose underlying operations are continuous with respect to its topology. Definition. A topological group G is a group with an underlying topology such that: the … Continue reading

## Topology: Separation Axioms

Motivation The separation axioms attempt to answer the following. Question. Given a topological space X, how far is it from being metrisable? We had a hint earlier: all metric spaces are Hausdorff, i.e. distinct points can be separated by two … Continue reading