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The following question is question 9.8 from Miller's paper ``Some interesting problems '':

Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, D_{n+1} \subseteq D_n$, then does $\bigcap_{n}D_n$ have the fixed point property?

As it is stated in the paper, the problem dates from the 1920's and was discussed by Kuratowski, Mazurkiewicz, and Knaster.

I wonder to know the status the problem, and if any progress or partial results are obtained about it.

I also remember, when I was passing undergraduate topology course, our teacher stated a very general open problem about the fixed point property of some subsets of the plain, from which the above could follow easily. I do not remember it correctly, so it would be helpful if someone states it, and gives references related to it (the problem was roughly something like: If $X$ is a compact connected subset of the plane, whose complement is also connected, maybe with some extra assumptions on $X$, then $X$ has the fixed point property).

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    $\begingroup$ I think the tag algebraic topology is more convenient. $\endgroup$ – Rahman. M May 18 '15 at 8:31
  • $\begingroup$ @YCor, Could you please describe the relation between geometric topology and this problem? $\endgroup$ – Rahman. M May 18 '15 at 10:24
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    $\begingroup$ @Rahman.M It's also not, properly speaking, algebraic topology, which is the study of algebraic invariants of topological space, and there is nothing algebraic in the question either. But both algebraic or geometric methods could be relevant to its solution, and both readers of either tag could be interested in the question. In practice, geometric topology is often associated to "low-dimensional topology" but the latter tag does not exist, so I viewed "geometric topology" as a field encompassing it. $\endgroup$ – YCor May 18 '15 at 10:54
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A reference concerning the fixed point property is the book "Open problems in Topology II" from 2007. See p263 ff (accessible from the preview) for a review of several problems concerning continua in the plane, contained in the article of Hagopian, "An update on the elusive fixed point property". In short there are tree-like continua that do not have the fixed point property but it is not known if they can be embedded in the plane. The problem of intersecting disks appears as Bing's problem 3 (from Bing's article, "The elusive fixed point property", 1969). I don't know if it is still open (it was in 2007).

Note that a plane continuum is the intersection of a decreasing sequence of disks iff it does not separate the plane. The question concerning plane continua you are refering to is probably the following (and would follow from a positive answer to your question).

Question. If $C$ is a continuum in the plane which does not separate the plane, does each homeomorphism of $C$ into itself leave some point fixed?

Edit: this question is solved affirmatively if $C$ is arcwise connected by Hagopian (1971).

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  • $\begingroup$ Thank you for your nice answer. Would you please give a reference for Hagopian's result? $\endgroup$ – Mohammad Golshani Jun 5 '15 at 6:11
  • $\begingroup$ I think you mean "does not separate the plane" in the Question. $\endgroup$ – Lasse Rempe-Gillen Jun 5 '15 at 7:54
  • $\begingroup$ The reference is: Hagopian, Charles L. A fixed point theorem for plane continua. Bull. Amer. Math. Soc. 77 1971 351–354. Now accessible at: projecteuclid.org/euclid.bams/1183532807 $\endgroup$ – YCor Jun 12 '15 at 20:50

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