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If $(X,\tau)$ is a topological space, let $\text{Im}(X)$ denote the collection of subsets $S$ of $X$ such that there is a continuous function $f:X\to X$ with $\text{im}(f) = S$.

Is there a space $(X,\tau)$ with $|X| > 1$ and with the following properties?

  1. The identity map $\text{id}_X$ is the only continous surjective map $f:X\to X$, and
  2. the partially ordered set $(\text{Im}(X)\setminus \{X\}, \subseteq)$ has no maximal elements.
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    $\begingroup$ The one-point space seems to work. Probably that isn't what you had in mind. $\endgroup$ – Joel David Hamkins Jan 2 '18 at 13:31
  • $\begingroup$ Right, thanks, I'll exclude that in the post! $\endgroup$ – Dominic van der Zypen Jan 2 '18 at 13:43
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A positive answer to this problem is given by the known answers to the following problem posed by de Groot in New Scottish book.

Problem 393 (de Groot; 28 May, 1958). Does there exist a (plane) continuum which does not admit any continuous map into itself except trivial ones.

This problem has been resolved by Cook (and Maćkowiak). The (historical) information on the solutions of de Groot' Problem 393 can be found on pages 308-309 of Mauldin's "The Scottish Book", New Edition of 2015. Mauldin writes that Knaster posed the same problem on the Topological Seminar in Warsaw University on October 22, 1930. So, the question of Dominic van der Zypen is in fact, very old.

It was answered by Cook (and Maćkowiak) who constructed 1-dimensional (planar) continuum $X$ whose any self-map is either the identity or constant.

It is clear that Cook's (or Maćkowiak's) continuum has the two properties required in OP of Dominic van der Zypen.

Remark. Topological space $X$ is called strongly rigid if each continuous self-map of $X$ is either constant or the identity. Many examples of (non-metrizable) strongly rigid spaces are constructed in this paper of Kannan and Rajagopalan.

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  • $\begingroup$ Very nice, Taras! - Interestingly, this is the 2nd time deGroot has been indirectly helpful in the context of a question I have been thinking about. I was busy during my PhD (around 2002) with the question whether every compact topology is contained in a maximal one. With H.P.Kuenzi we were able to resolve some special cases in the positive, but then M.M.Kovar resolved the whole question in the positive -- with the deGroot dual. $\endgroup$ – Dominic van der Zypen Feb 13 '18 at 13:42

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