Groups and their Basic Properties
Essentially, a group is a set endowed with a very basic structure. This structure is enforced by an operation which governs how the elements in the group interact with each other.
Bases for Topologies
Essentially, a basis is a 'small' collection of open sets from which every open set can be easily generated. It is often useful to talk about the topology generated by a specific basis, since many facts about a topology can be gleaned by studying one of its bases.
Mathematical Induction
Before my next post on bases for topologies, I need to introduce a proof technique that I haven't used so far.
A First Look at Topological Spaces
There are also many circumstances in which we care about the shape of a space but couldn't care less about distances. For instance, a famous puzzle that influenced the development of the entire field of topology is the problem of the **Seven Bridges of Königsberg**.
Metric Spaces (2)
Looking back through my first post about metric spaces, it occurred to me that I should probably have emphasized a few things that could be a bit confusing, so let me address those first before pressing forward.
Metric Spaces (1)
Taken by themselves, sets do not have much structure to them. They are essentially barren wastelands with no relationships at all between their elements. In this post we will remedy that by defining a way to add a measure of proximity to the points in a set.
Set Theory
Everything in mathematics is built from sets. Even objects such as functions and arithmetic operations like addition are formally defined in terms of sets, although you would likely never expect it.