Constructing an Inverse Sequence for a Topological Space

Question

There is a ton of literature out there about inverse limits of inverse sequences of topological spaces with continuous bonding maps. Arguably the most studied kinds are those inverse sequences whose factor spaces are all homeomorphic to $[0,1]$. Sometimes the bonding maps are the same, sometimes they can vary. My problems with them, however, is the following: intuition and construction.

First of all, many papers and books on this topic claim that they are a powerful tool since they can express extremely pathological and complicated spaces in terms of simpler ones (like the unit interval mentioned above). However, I disagree. For one, it gives no way of knowing intuitively what the space will be or look like from the definition of an inverse limit. Second, while inverse limits do have their analytic purpose, it seems like the most complicated tool one can build to achieve anything.

There are so many papers on inverse limits, most of which are just more theorems about them but almost never showing examples of spaces expressed as inverse limits. Even when they do give examples, they only state them as such and never show how one arrives at that conclusion. For example, the famous Bucket Handle continuum, which is indecomposable and non-locally connected at every point, can be expressed as an inverse limit of an inverse system whose factor spaces are all $[0,1]$ and whose bonding maps are the tent map with slope $2$ and critical point $1/2$. How is it that anyone ever came to such a conclusion? How can one start with a continuum (compact connected metric space) and come up with a way to express it as an inverse limit? How does one go about constructing an inverse system for any given space?

UPDATE: After doing some research, I found what appears to be a very useful source, at least for chainable continua. Inverse Limits, From Continua to Chaos by W.T. Ingram and William S. Mahavier, offers an algorithm for constructing a map from $[0,1]$ onto itself which is piecewise linear and non-constant on any subinterval, based on two initial ``taut" chains, one of which strongly refines the other. It's intuitive, even though the written construction is pretty technical.

Answer 1

In general, it's quite hard to start with a continuum and construct an inverse limit representation. That's why most such constructions are celebrated, as they're super useful and generally not trivial. In my particular field (tiling spaces), inverse limit representations are useful for calculating topological invariants for instance, which would otherwise be essentially impossible to determine. The paper that first showed these spaces could be constructed (in a nice way) as inverse limits is now one of the most cited in the field. There are also many other reasons why inverse limits are useful for studying a pathological topological space.

Answer 2

Inverse limits of the transfinite type (indexed by $\omega_1$, e.g.) are used in some famous examples like Kunen's compact $L$-space and Fedorchuks's compact $S$-space without convergent sequences, and others. The long "length" of these sequence allows us to "build in" properties of the eventual limit. Studying proof of such examples might help one to build intuition. Another advantage of such "spectra" (as such inverse limits are called in Russian papers, sometimes) is that a map between their limits is (under some conditions) always a limit map of a sequence of maps on the composing spaces (Sčepin's theorem), so we can control things like homogeneity or homeomorphic-ness of those limits.

As a general theorem, one can show that any compact space of weight $\aleph_1$ is the limit of an inverse system (not always a sequence) of compact metrisable spaces. (this follows from facts about embedding theorems into products and how products and some subspaces can always be seen as limits of inverse systems) Such general facts show, I think, that inverse limits are not "special", but that it is indeed hard to visualise what the limit of a given system looks like. The inverse sequences (countable) of intervals, came to be studied because people were interested in the orbits of iterated maps on intervals, I think. The herediatarily indecomposable continua were first constructed as the result of other iterative processes in the plane, intersections of chains that were decreasing in size etc. Maybe look at the original papers and compare them to the inverse limit view of them.