In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. This seems warranted by some of the nice properties of ANRs:

Every ANR has the homotopy type of a CW complex.

Every ANR is locally contractible, and as a partial converse, any locally contractible finite-dimensional metric space is an ANR.

But there are also important infinite-dimensional examples of ANR's:

The Hilbert cube is an ANR.

Many function spaces are ANRs.

This leaves me with some

**Questions:**

What is a good example of a finite-dimensional ANR which is not a CW complex?

Are (finite-dimensional) ANRs an appropriate setting to study either (a) fractals (wikipedia seems to define a fractal to be a subset of Euclidean space whose topological and Hausdorff dimensions differ) or (b) the limit sets of dynamical systems on CW complexes? I think my sense is that both (a) and (b) are generally

*wilder*than ANRs, but I'm not really sure -- perhaps there's some overlap but no strict containments?Do ANRs admit some kind of "generalized cell structure" like CW complexes do? Or is there some other sense in which ANRs can be "classified"? Is there at least a "classification" of what ANRs can look like locally?

Less precisely, my feeling is that when somebody says "Let $X$ be a CW complex", I sort of know what they mean. But when somebody says "Let $X$ be an ANR", I don't -- I don't know what to think of as a "typical example", nor do I know what kinds of "typical pathologies" to watch out for. It would be nice if there were a book out there entirely devoted to the topology of ANRs ~~ but surprisingly I haven't been able to find one ~~. I found a book Theory of Retracts by Sze-Tsen Hu, but I haven't yet found an example in it of a finite-dimensional ANR which is not a CW complex.

**EDIT:**

Another reference is Borsuk's

*The Theory of Retracts*. This contains more examples in the later chapters, though I'm still struggling to piece together a coherent picture of the diversity of ANRs.An important piece of context regarding (1): according to Thm V.10.1 of Borsuk, the (compact) finite-dimensional ANRs coincide with the retracts of (finite) polyhedra. Thus in finite dimensions, the question is:

*How wild can an idempotent on a polyhedron be?*.

localproperty (and the dogbone space seems like it was cooked up to have interestingglobalproperties). $\endgroup$ – Tim Campion Jun 29 '19 at 16:33