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The only reference I'm familiar with that deals with arc accessibility in the plane is Wilder's Topology of Manifolds, which is getting close to 100 years old. There is a book on Cluster Sets which covers it up to what's typically needed in complex analysis; there's also a monograph on some of Bell's work, but it's not really didactic. There have been plenty of refinements and applications of accessibility since the Wilder book in particular, but I had trouble finding a serious treatment for topologists, rather than complex analysts.

Anyone know a good one?

John Samples
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  • Maybe Continuum Theory. An Introduction by Sam B. Nadler (1992)? – Dave L. Renfro Feb 07 '21 at 02:06
  • You'd think so, but it's not covered at all (except for a couple exercises). Very annoying omission. – John Samples Feb 07 '21 at 02:08
  • It should be in Moise's book. – Moishe Kohan Feb 07 '21 at 03:40
  • @MoisheKohan You mean the Dimension 2/3 book? That one also just has some exercises here and there. – John Samples Feb 07 '21 at 04:41
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    For more about arc accessibility in the sense of cluster set theory, some of which might be of interest to you (especially when continuous functions or arbitrary functions are involved), see the google search "ambiguous point" + "Bagemihl" + "cluster", although not everything even marginally well known on the topic appears to show up with this search, such as this paper. – Dave L. Renfro Feb 07 '21 at 09:17
  • Right, I meant that book. See proofs in chapter 9. Bing also has it in his book "The geometry and topology of 3-manifolds", page 28 (with proofs). If this is not enough, modify the question and explain what exactly are you looking for. – Moishe Kohan Feb 07 '21 at 10:51
  • Is there a newer edition of the Moise? In my copy, "arcwise accessible" isn't even defined until the exercises in chapter 9, which is the only place the phrase shows up. I'll look at Bing. The request is for a thorough treatment of the theory younger than World War II basically xD – John Samples Feb 07 '21 at 11:52
  • @DaveL.Renfro Ah ok, this looks like a good lead. I'm starting to get suspicious that there just isn't really a book that deals with it. I checked the Bing, there's basically nothing. Daverman has nothing and Sakai's books also have nothing. I totally forgot about the non-Nadler continuum theory books that came out the past 10 years or so; also need to look in Kuratowski/Arkhangelskii. – John Samples Feb 07 '21 at 12:01
  • Both Bing and Moise prove that each Jordan arc in the plane is arcwise-accessible. What are the other results you are interested in? People from Complex Analysis are naturally interested in this since it relates to the extension problem, so there is much more developed theory of prime ends. – Moishe Kohan Feb 07 '21 at 15:09
  • There are lots of refinements of accessibility by arcs with certain properties, or by other sets, or how the arcs can or can't foliate the ambient space, their analytic properties. As mentiond there's a theory of cluster sets/prime ends/cross cuts that often is topological. There are a lot of important results concerning Peano Continua, especially boundary curves, and the topological/analytic properties of the types of domain they bound determining things about accessibility (e.g. property S). There is 1/2-sided accessibility . . . would like to know about specific curves like dendroids etc – John Samples Feb 07 '21 at 19:21
  • Especially a book covering some of the work of Hagopian would be really nice, but that's almost certainly asking too much. Then there's the theory of channels which is considered fairly important and maybe the best technique available atm for attacking the plane fpp/cellular fpp. – John Samples Feb 07 '21 at 19:24

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