2. Topology
What do we mean with topology? Wikipedia describes topology as follows:
In mathematics, topology … is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
What does this even mean?
 Coordinate invariance

Topology considers how things are connected, not the actual coordinates. The ellipses below are all topologically equal; it does not matter if they are rotated or not.
 Deformation invariance

The geometric shape of an object can be deformed by stretching or bending it, as long as we don’t create new holes or remove existing ones. The letters
A
written in two different fonts below are topologically the same. The letterB
is different because we cannot deform a letterA
to create a letterB
without creating a new hole.
 Compressed representation

Consider a circle consisting of an infinite number of points. We can represent this same circle with one triangle consisting of only 3 points; they are topologically equivalent.
Image source: Ayasdi white paper "Deep Dive: Topological Data Analysis"
A running joke tells of a topologist who is not able to distinguish a mug from a donut. Indeed, these are equivalent. If you start making the bottom of the mug thicker and thicker, you’ll be working towards a donut (torus) shape. Both have only 1 hole: the handle, and the hole in the donut.
2.1. Topology vs geometry vs algebra
Consider the idea of a "circle".
In geometry, we talk about its curvature, width, rotational symmetry, etc.
In topology, we look at the circle as being made from flexible material: we can stretch and deform it, but should not poke hole in it or break it.
In algebra, we consider the circle as a collection of points and a rule to connect them. Here, we have a notion of "nearness" of points. This will become important later when we talk about "algebraic topology".
To summarise: a donut and a mug have a different geometry but the same topology, whereas a sieve and a plate have a similar geometry but a different topology (i.e. number of holes).
The image below shows topological equivalence across a wide number of shapes. It shows that a disk is equivalent to a singular point or a hollow sphere with a hole in it. Also, a number 8
is equivalent to two circles connected with a line, etc.
Source: Singh G et al (2008) Topological analysis of population activity in visual cortex. Journal of Vision, 8(11), 118
2.2. From data to topology
As we’re working with datapoints (often in highdimensional space) and topology (in terms of torus shapes, circles, etc), how do we marry these two together?
In topological data analysis, we convert the datapoints into a network (actually simplicial complexes, but we’ll go into that later). There are different ways for doing this, depending on whether it is for visualisation purposes (mapper), or for analysis (persistent homology).
We’ll look further in both of them in the following sections. Things that we are particularly interested in, are flares and holes.