One of my little dreams is to make and give a workshop about topology for artists, walkers and outher outsiders. It might be called: The topological toolkit. What you didn't know about spaces. Or maybe: Topology as philosophy of space. These are preliminary notes: YouTube topology, Styles of topology, Spaces and maps, Translation, Words in linguistic space. I try to post small updates often, to keep myself working on this.
I read a lot of discussions on topology. Below is a summary of what I learned, structured as a workshop outline. See the list of sources at the end.
The nature of space
Topology studies the nature of the space itself. Does it have holes? Is it close together (compact) or spread out? This allows higher math to define calculus over strange spaces.
- Think about the difference between the sphere and the torus.
- Discuss the possibility of living in finite but unbounded space (asteroids game).
(Point set) Topology as basic theory
It's like calculus, linear algebra, set theory or most other fields of mathematics that are taught in college or the first year of graduate school. You have to know them because you'll be using them in whatever you will be doing as a mathematician. You probably won't be doing research within one of these fields, but they're all used in current research.
I’m not aware of any applications of pure point set topology. Usually, applications involve more complicated topological structures, like manifolds or knots. However, if you want to really understand these more complicated structures, you need to understand the more basic theory, particularly because point set topology shows you just how unintuitive general topology is. In the general setting, there are all sorts of weird counterexamples that don’t feel like they should exist, but you need to be aware of these if you want to have the right intuition of how much information you need to be able to solve various problems.
- Discuss the extremely basic concepts introduced in topology, that demonstrate that "things could have been totally different and very weird" (like the multiverse).
- Try to show some weird spaces, very different from ours (maybe using separation properties).
Structures on a space
Point set topology is the basic study of topological spaces. A topological space is sort of the minimal amount of structure that you need in order to be able to talk about convergence, connectedness, and continuous functions.
Most modern mathematics concerns "sets with structure" of various sorts. The problem is that the structure of "topological space" by itself isn't really strong enough to produce much interesting mathematics--topological spaces by themselves are too wild to make many interesting statements that hold for all topological spaces. Point-set topology studies potentially pathological topological spaces from an essentially set-theoretic point of view.
- Think about the building blocks of our daily space.
- Discuss bases.
- Discuss the standard (open balls) and the discrete (points) topology on a space.
- Discuss how different topologies lead to different and weird properties of a space (Sorgenfrey line is much too complicated for workshop).
- Try to present a few "wild" spaces (if possible).
Geometry without a metric
Topology seems to have arisen to answer the question "What is geometry without a metric, without a way of measuring distance?" It's a study of geometry where there isn't a metric -- a way to measure the distance between two points. Several lines of thought from different areas converged into questions about the nature of the space itself.
- Think about the different practical metrics in our daily (Euclidean) space.
- Discuss measuring distance in kilometres versus measuring distance in time or in difficulty.
- Discuss the difference between car, bicycle or walk distance.
- Discuss manhattan-block distance (pixels).
- Discus distance through and on a sphere.
- Discuss the space of music or literary styles.
In the broadest sense, topology formalizes the notions of nearness and continuity. Reality appears to behave in a mostly continuous manner, at least at the scale on which humans and human intuition evolved. We tend to think of things in a continuous manner, we tend to interact with reality in a continuous manner, and now we have topology to formalize all of these continuous intuitions and interactions.
It is like geometry in that studies space, but it does not worry about the nuisances of angles and distances (even if metric spaces are touched on, which do use the latter), instead simply speaking of “closeness” (adjacency).
- Think about continuity and discontinuity in our daily surroundings.
- Discuss the absence of infinities, and infinite gradients, in daily experience.
Open and closed
Topology (point-set or general topology, that is) fundamentally captures the idea of continuity in a general setting, via open sets (or closed sets, if you like) which remain open after pullback.
- Think about open and closed sets.
- Discuss the openness or closedness of concepts like "marxist" or "city centre".
Topology is also cool in its own right. It is fascinating that we can rigorously consider objects continuously deformable into each other to be equivalent (c.f., homeomorphism & homotopy equivalence) and get something out of that. I am sure you have heard of the donut and the coffee cup, which gets the vague idea across.
- Think about mappings between different domains.
- Think about difference between map and reality.
Countable and uncountable
- Countable and uncountable sets (maybe mention Cantor and infinities).
- You can cover an uncountable space with a countable set of open sets (I think).
- How to make this practical? I find Q versus R fascinating, but is it explainable to outsiders?
(Point set) Topology as a mature subject
Point-set topology is a “dead” field the same reason classical real variables or elliptic integrals are: the subject has been studied extensively, the hidden nuggets have been extracted, and further study of it is largely considered uninteresting. Everything in it has already been studied and proven.
Point set topology is basically a dead field, and is really not a popular research topic. For most interesting problems that use topology, it is necessary to assume spaces are nice enough that stronger techniques--such as those from algebraic topology, differential topology, or functional analysis--become available. Certainly point set topology forms a foundation for a lot of really interesting mathematics, but it isn't really studied much for its own sake anymore.
- Since 1900 mathematicians have thought about the structures and properties of space. This was done to produce a solid theoretical underpinning of "normal" mathematics.
- Topology is more philosophical than practical. It presents us with puzzles and makes us think.
For further study
You must be comfortable with concepts like compactness, connectedness and homeomorphisms. You also need partitions of unity, subordinate to an open cover (very important!) and for this you have to be familiar with (something like) the smooth Urysohn lemma. Finally, some understanding of quotient topology will be a plus, since it gives many important examples (Klein bottle, Projective space).
- Find the connection between point set topology and the weird structures of the klein bottle or the projective plane. I haven't found that yet.
Importance of topology:
For later research: