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I've noticed many topological properties have a fundamental object for which we have to put in some elbow grease to establish it from scratch, after which there is a slew of theorems allowing us to build our way to checking the property for various complicated objects. Here are some examples so that you know what I mean by properties, objects, and building.

Connected: The fundamental object is $[0,1].$ A special sup trick is needed and you gotta get your hands dirty. After proving $[0,1]$ is connected, you can prove many complicated spaces are connected. The implication "path connected implies connected" is a direct consequence and allows us to show the connectedness of large classes like convex spaces, various subsets of $\mathbb{R}^n,$ and various subsets of metric spaces.

Compact: The fundamental object is $[0,1]$ again. The classic contradiction proof of the compactness of $[0,1]$ goes to the lowest levels of soups. The fact that closed subsets of compact sets are compact and products of compact spaces are compact gets you really far, but you have to start somewhere. If you track down the root of many proofs, it's all thanks to $[0,1].$

Fundamental Group: The fundamental object is $S^1.$ To establish $\pi_1(S^1) = \mathbb{Z}$ requires setting up covering spaces, covering maps, and many auxiliary lemmas that look like they came out of nowhere. But once you know $S^1,$ you can work your way up via products and the Seifert-Van Kampen Theorem. If you track down the computation of many fundamental groups, it's all thanks to $S^1.$

Complete: The fundamental object is $\mathbb{R}.$ That $\mathbb{R}$ is complete is either by definition or as a consequence of an alternate definition once you show equivalence of definitions. The completeness of finite dimensional spaces (whether the dimension is vector space dimension or something else) comes down to the completeness of $\mathbb{R}.$ Even in the realm of Banach spaces, you can't escape real numbers. To prove a Cauchy series of vectors converges, you often use that the vector sequence is Cauchy to get a Cauchy series of real or complex numbers, and then use the convergence of that resulting series to go back to the vectors. If you level up to Banach algebras, now you show completeness by reducing a sequence of operators to a sequence of vectors. The root case is $\mathbb{R}$ or $\mathbb{C},$ and for $\mathbb{C}$ the root is $\mathbb{R}.$

Is there a way to rigorously define what a (fundamental object, property) pair is in the spirit of these examples, perhaps with category theory? Are there other examples of properties and corresponding objects (I've made this a list question)?

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  • I believe a more suitable fundamental object for compactness is a singleton set, but this is motivated purely by my interests – Carlyle Nov 30 '23 at 21:06
  • Besides that, I wonder if a fundamental object is the right idea, if we pretend that for some property we know only its fundamental object, I wouldn't feel like I know anything about the property. If however we knew of a collection of relations such that any two objects that are related by one of these relations have the same valuation with respect to our property, then I feel like I know more about the property, for example with compactness we have "is a closed subset of" as one of the relations, and also "is a product of", but I realise this is not what you are looking for – Carlyle Nov 30 '23 at 21:13
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    It's strange thing to say that the fundamental object for $\pi_1$ is $S^1$, given that it literally is defined using $S^1$. The same applies to metric spaces (including complete), which have $\mathbb{R}$ baked into definition. So of course there is no escape. Also, its an exaggeration that the fundamental object for compactness is $[0,1]$. Bolzano-Weirstrass, and equivalence of sequential compactness with compactness for metric spaces have little to do with the interval. – freakish Nov 30 '23 at 23:17
  • Also note that $S^1$ itself cannot be constructed without reals. Concluding: what you seem to say is that real numbers are important in maths. As a building bricks for the unit interval, complex numbers, spheres, etc. Indeed, they are, and indeed real numbers have special place in maths, unlike any other space. – freakish Nov 30 '23 at 23:44

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As noted in the comments, the list of these "fundamental objects" is questionable. However, we can make the observation that $[0,1]$ is the "fundamental compact space" maybe precise by saying that it is an injective cogenerator in the category of compact Hausdorff spaces. But this theme does not generalize. For example, in the category of abelian groups, $\mathbb{Q}/\mathbb{Z}$ is an injective cogenerator, but it is more like (one half of) the "fundamental divisible abelian group", not really like the fundamental abelian group. This role is taken by $\mathbb{Z}$, the projective generator.

Martin Brandenburg
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