General Topology – Visualizing the Hierarchy of Mathematical Spaces

functional-analysisgeneral-topologygeometryvisualization

I was inspired by this flowchart of mathematical sets and wanted to try and visualize it, since I internalize math best in that way. This is what I've come up with so far:

Version 1 (old diagram)

Version 2:
enter image description here

Is there anything that I'm missing, or that is incorrectly marked? For example, where exactly should I insert a box for Fréchet Spaces? And, is it safe to say that Normed Vector Spaces are a proper subset of the intersection between Locally Convex Spaces and Metric Spaces (or is it the entire intersection?)

Edit:
Thank you, everyone, for your input. Obviously no single diagram is going to encapsulate the entirety of functional analysis, geometry, and topology (not to mention the myriad of algebraic structures I've ignored, as some of you have pointed out.) As someone who does a lot of analysis, I would often find myself going back to Wikipedia or my textbooks to re-read the definitions of the various spaces and sets I am working with. I just wanted something that could help me keep a lot of these ideas straight in my head; and was pretty and useful to glance at. I think I've settled on my final version (for now.) In summary, here is a quick bullet list of the labeled components of the diagram:

  • Topological Spaces: sets with a notion of what is "open" and "closed".
  • Vector Spaces: sets with operations of "addition" and "(scalar) multiplication".
  • Topological Vector Spaces: "addition" and "multiplication" are continuous in the topology.
  • Metric Spaces: sets that come with a way to measure the "distance" between two points, called a metric; the topology is generated by this metric.
  • Locally Convex Spaces: sets where the topology is generated by translations of "balls" (balanced, absorbent, convex sets); do not necessarily have a notion of "distance".
  • Normed Vector Spaces: sets where the topology is generated by a norm, which in some sense is the measure of a vector's "length". A norm can always generate a metric (measure the "length" of the difference of two vectors), and every normed space is also locally convex.
  • Fréchet Spaces: a set where the topology is generated by a translation-invariant metric; this metric doesn't necessarily have to come from a norm. All Fréchet spaces are complete metric spaces (meaning that if elements of a sequence get arbitrarily "close", then the sequence must converge to an element already in the space.)
  • Banach Spaces: a set that is a complete metric space, where the metric is defined in terms of a norm.
  • Inner Product Spaces: sets with a way to measure "angles" between vectors, called an inner product. An inner product can always generate a norm, but the space may or may not be complete with respect to this norm.
  • Hilbert Spaces: an inner product space that is complete with respect to this induced norm. Any inner product space that is incomplete (called a "pre-Hilbert Space") can be completed to a Hilbert space.
  • Manifold: a set with a topology that locally "looks like" Euclidean space. Any manifold can be turned into a metric space.

Best Answer

My advice is to place a lot more landmarks like $\mathbb R^n$. Ideally, every area should have at least one point in it, which will serve to prove that the area really belongs there. It will also clarify what the relationships really mean. For example, all manifolds are metrizable, but not uniquely. So if you want "manifolds" to extend outside of "metric spaces", then you should add a landmark like $S^1$ and then, in a list of landmarks below the diagram, explain why it's there:

$S^1$ denotes the circle as a topological space. It is a manifold. It is not homeomorphic to any real vector space, since it is compact. It is metrizable, like all manifolds, but it doesn't come equipped with any particular metric.

Speaking of which, manifolds have a finite dimension, which is a topological invariant. So if a real manifold does have a real vector space structure, then it is a finite-dimensional vector space, and that may make it difficult to draw meaningful distinctions within all the little slices in the manifold box. Again, depending on what you really mean, you might be able to justify those slices, so I'm not going to say that they're wrong. Trying to place landmarks in there will force you to decide what you want them to mean.

Once you go through enough examples, then you can summarize the meanings in a preface to the diagram:

This diagram depicts X. One box is placed entirely inside another box if either Y or (when it makes sense) Z.

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