Topology is the study of connectedness of objects in a mathematical space. A mathematical space equipped with a topology is called a topological space. For a podcast discussion on this, see episode 133.
Topology is more general than geometry, because geometry is concerned with distances and angles, whereas topology is only concerned with the connections between points. Topological objects can be stretched and skewed and remain the same mathematically, whereas geometric objects are considered different when changed in that way.
For example, a 2-dimensional surface can come in many topological forms: A sphere, a torus (doughnut), or even a patch with a round edge. These are all topologically distinct, but they are topologically equivalent to the following:
- The sphere is equivalent to a cube, or any closed shape. Even a cup can be considered a sphere with an indent.
- The torus is equivalent to any shape with a single hole through it. One example is a coffee cup, with the hole being the handle.
- The round patch is equivalent to a 2d patch of any shape, so long as there are no holes in it.
The only way to fundamentally change a topological object is to “glue” points together (add connections where none existed before) or to “tear” then apart (remove connections).
On the discrete side, a network topology is just a directed graph, where each point has a distinct set of “neighbors” connected to that point. Network topology has many important use cases in engineering and communication networks.
General topology usually refers to some kind of continuum or dense space, where points don’t have explicit neighbors. For example, a point on the real number line, or a point in a 3 dimensional space cannot be said to be connected to another distinct point because there are always points in between.
In this case, general topology calls for neighborhood space where each point is equipped with a neighborhood system instead of simply a set of neighbors. Typically, each point has many “neighborhoods” which are sets that completely surround and contain the point. If one wants to approach the point from afar, every single neighborhood of that point will have to be entered before reaching the point.
This allows for a general mathematical treatment of continuity and limits.