11.1 Topology

According to the Cambridge English Dictionary, the word “topology” means

the way the parts of something are organized or connected.

From a mathematical perspective, topology studies the local properties of structures and spaces.

In machine learning, we are often interested in global properties like minima and maxima but only have local tools to search for them. One example is the derivative of functions. Derivatives describe the slope of the tangent plane, and as Figure 11.1 illustrates, this doesn’t change if the function is modified away from the point where the derivative is taken.

Figure 11.1: The derivatives of two functions are equal if the functions are equal in any small interval around the point

In mathematics, local properties are handled in terms of sequences and neighborhoods. We learned about sequences in the last chapter, and now we will tackle the subject of neighborhoods.

We are going to focus on three fundamental aspects...