0

Outside mathematics, the way I have heard the word topology being used is to describe the shape of, say, terrain: "Topology of the route we are about to hike is tough!". Given that the mathematical topology is just a collection of sets which satisfy certain conditions (empty set and unions & certain intersections belong to the collection), is there an "easy" correspondence between the usage of the word "topology" outside mathematics, or is it just one of those things where we have taken a certain word in mathematics to describe the rules we are interested in in a more "intuitive" way (like open/closed sets)?

Epsilon Away
  • 954
  • Topology in mathematics does describe shape in a sense. It is not obvious from the abstract definition, but it becomes clear studying the topology of geometrical objects like surfaces. – Javi Jan 13 '22 at 09:39
  • 2
    Could this be a confusion between topology and topography? – Michael Burr Jan 13 '22 at 09:40
  • 1
    "Topology of the route we are about to hike is tough"- I do not think it is a proper English sentence. – markvs Jan 13 '22 at 09:50

1 Answers1

1

Outside mathematics, the way I have heard the word topology being used is to describe the shape of, say, terrain: "Topology of the route we are about to hike is tough!".

I'm pretty sure that whoever said that he actually meant topography. It is a common mistake.

That being said, there is some grain of truth here. Both topology and topography describe shapes. Topography studies features of terrain, from geoscience point of view. While topology studies features of shapes and their distortions, from mathematical point of view.

So you are right that topology has quite abstract definition via collection of open subsets under certain rules. But this definition does not come out of thin air. It actually started from studying some classical shapes: lines, triangles, spheres. It is often the case, that mathematicians don't want to study each item separately, so they tend to search for common denominator to study. What is the difference between line and triangle? What is similar between line and circle? Depending on the goal of course. And one of the ideas that turned out to catch those similarities and differences quite well is the abstract topology. Moreover it turned out to be useful in other branches of maths.

So for example the standard sphere. Its topology says quite a lot about its shape: it is closed, it is bounded, it has a single hole (whatever that means). By studying it futher (i.e. homotopy) it can be shown that $n$-dimensional sphere cannot be deformed into a $m$-dimensional sphere (when $n\neq m$). And no sphere can be deformed to a point. All of that from this simple abstract setup. Of course topology does not catch all features. Circle of radius $2$ is the same as circle of radius $1$ from topological point of view. For that you would need some other (geometrical) approach.

freakish
  • 42,851