# 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

## Topology: Locally Connected and Locally Path-Connected Spaces

Locally Connected Spaces Recall that each topological space X is the set-theoretic disjoint union of its connected components, but in general (e.g. for X=Q) fails to be the topological disjoint union. The problem is that the connected components in general aren’t open … Continue reading

## Topology: Path-Connected Spaces

A related notion of connectedness is this: Definition. A path on a topological space X is a continuous map The path is said to connect x and y in X if f(0)=x and f(1)=y. X is said to be path-connected if any two points … Continue reading

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## Topology: Connected Spaces

Let X be a topological space. Recall that if U is a clopen (i.e. open and closed) subset of X, then X is the topological disjoint union of U and X–U. Hence, if we assume X cannot be decomposed any further, there’re no non-trivial clopen subsets of X. … Continue reading

## Topology: One-Point Compactification and Locally Compact Spaces

Compactifications There’re lots of similarities between completeness and compactness, beyond the superficial resemblance of the words. For example, a closed subset of a compact (resp. complete) space is also compact (resp. complete). Two differences though: compactness is a topological concept … Continue reading