General Topology

General topology, also called point set topology, is the study of properties of sets and functions that relate to notions of continuity, connectedness, and various abstract conceptions of "closeness." Topology notably dispenses with the notion of numerical distance in order to handle these concepts. Instead, topology generalizes more concrete concepts from geometry and analysis to more abstract concepts involving sets. The fact that topology is a generalization of a more familiar concept means that properly motivating topology involves explaining its more concrete predecessors.

One such familiar topic comes that comes from analysis is continuity. Informally speaking, a function in the Cartesian plane is continuous if we can draw its graph without picking up our pencil. A more formal definition of continuity for functions of real numbers involves the use of a metric, which is a function that itself formalizes the concept of distance between points. This more formal definition says that a function is continuous if, as the difference between two points $f(x_0)$ and $f(x_1)$ in its codomain gets very small, the difference between the values $x_0$ and $x_1$ in the domain also becomes very small. Topology asks if we can generalize this notion of continuity without having to rely on the notion of numerical distances.

Another topological concept is that of connectedness, which formalizes whether a set can be broken out into at least two separate chunks that "don't touch." Again, the notion of "touching" is one of more general "closeness" that doesn't exactly depend on concrete distances. For example, two circles either do not touch at all, touch exactly at their perimeters, or overlap. Topology asks whether we can formalize these ideas of not touching, touching just-so, and overlapping, without needing to know the concrete numerical extent to which they are separated or overlap.

A more geometric understanding of topology is that it studies that properties of curves and surfaces that don't change when you stretch or smush them around but do change if you manage to tear or split up the surface. For example, consider a spherical stress ball. It doesn't stop being smooth (i.e. continuous) when you squish it around. Likewise, tires and donuts both have holes inside inside them, even though the rest of their surface geometry can be quite different. While you can conceptually stretch a donut into a tire, you can't stretch a stress ball into a donut without adding a hole somehow or merging two parts of the surfaces together. The properties of a shape having a hole or remaining smooth under deformation are concepts that are formalized and generalized in topology.

The Topological Spaces sections go over basic definitions and concepts that form the foundation of higher order concepts. The Topologies section then introduces a number of examples of how to construct particular topologies. Subsequent sections make use of these examples to illustrate the concepts of closed sets, continuity, connectedness, compactness, and more.