## From Euler Characteristics to Cohomology (II)

Boundary Maps Here’s a brief recap of the previous article: we learnt that in refining a cell decomposition of an object M, we can, at each step, pick an i-dimensional cell and divide it in two. In this way, we … Continue reading

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## Elementary Module Theory (IV): Linear Algebra

Throughout this article, a general ring is denoted R while a division ring is denoted D. Dimension of a Vector Space First, let’s consider the dimension of a vector space V over D, denoted dim(V). If W is a subspace of V, we proved earlier that … 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