Tag Archives: universal properties

Topology: Complete Metric Spaces

Posted on February 21, 2013 by limsup

[ This article was updated on 8 Mar 13; the universal property is now in terms of Cauchy-continuous maps. ]  On an intuitive level, a complete metric space is one where there are “no gaps”. Formally, we have: Definition. A … Continue reading →

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Topology: Product Spaces (II)

Posted on February 16, 2013 by limsup

The Box Topology Following an earlier article on products of two topological spaces, we’ll now talk about a product of possibly infinitely many topological spaces. Suppose is a collection of topological spaces indexed by I, and we wish to define … Continue reading →

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Intermediate Group Theory (6)

Posted on October 10, 2012 by limsup

In this post, we’ll only focus on additive abelian groups. By additive, we mean the underlying group operation is denoted by +. The identity and inverse of x are denoted by 0 and –x respectively. Similarly, 2x+3y refers to x+x+y+y+y. Etc … Continue reading →

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Casual Introduction to Group Theory (6)

Posted on September 28, 2012 by limsup

Homomorphisms [ This post roughly corresponds to Chapter VI of the old blog. ] For sets, one considers functions f : S → T between them. For groups, one would like to consider only actions which respect the group operation. Definition.  Let G and … Continue reading →

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