## Geometrical versus Topological

NYC geographical map shown below is an example of geometrical representation. The distances between the boroughs of NYC are proportional to the actual distances. NYC subway diagram shown below is an example of topological representation. On this topological map the distances between the boroughs of NYC are NOT proportional to the actual distances. You can ignore the background shading and just focus on the nodes and the connecting lines. A topological map is known as a “graph” in mathematics.  Mathematical graphs model pairwise relations between objects. A mathematical graph is made up of vertices (nodes) which are connected by edges (lines, arcs). Topological maps lack scale.

Topology is a huge subject. It is much more than connections and graphs. Here’s few reminders from Wolfram MathWorld

“Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed. A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid.” –

“Topology can be used to abstract the inherent connectivity of objects while ignoring their detailed form.”

Topology can be divided into algebraic topology (which includes combinatorial topology), differential topology, and low-dimensional topology. The low-level language of topology, which is not really considered a separate “branch” of topology, is known as point-set topology.

Two objects can be very different geometrically but equivalent topologically. A coffee mug is topologically equivalent to a torus even though the two objects are very different geometrically. A torus is NOT topologically equivalent to a sphere because a torus cannot be deformed into a sphere. In the geometrical objects shown below count the number of holes. If the number of holes is the same then the geometrical objects are topologically equivalent. If you are curious about the relationship between Topology and Graph Theory you can read the comments here. 