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There are different versions of the Suslinian property for continua. One is called finitely Suslinian, a term I think I learned about from the OP - if I recall correctly, it asks that, for any eps>0, …

Here's a slightly more hands-on point of view, using just the Jordan curve theorem (although of course in the end it does come down to the same thing as Euler's formula somehow, as described by Alex.) …

Here is one possible elementary argument (somewhat inspired by my paper with Shen in the Monthly, "The exponential map is chaotic"), which avoids any mention of the classification of Fatou components, …

Let $f$ be the complex polynomial $f(z) = z^2 - 1$. Its Julia set $J(f)$ is the set of non-equicontinuity of the iterates; i.e., the set of points whose orbit under $f$ is not stable under perturbatio …

I am interested in a reference for the following fact (or a similar result). PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with …

I just stumbled upon this old question, and thought I would add a simple and natural example, which is $G_{\delta \sigma}$ but neither $G_{\delta}$ nor $F_{\sigma}$. Consider $f\colon \mathbb{C}\to\ma …

My question is as follows. QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f …

I believe the question whether $X$ can be chosen to be Hausdorff was left open by both existing answers. The solution is provided by the H-closed spaces of Henno Brandsma's comment. I shall answer the …

EDIT. As requested, I am extending the answer, including the relevant definitions. To do so, I have also re-arranged the answer somewhat. I am also now including the notions of surjective span and sem …

There are indeed quite a number of definitions of end-points, or of terminal points (the terms are sometimes used interchangeably, sometimes not) of a continuum $X$, as I discovered recently when I wa …

I know this is an old question that already has an excellent answer (although it may apply only to compact sets). However, let me respond to the original question, concerning the following results: …

In the broad sense in which you state your question, and if you are looking for rigorous results, then the answer is surely no. Keep in mind that the existence of the Lorenz attractor was only prove …

Let $K$ be the boundary of your circle domain $\Omega$. Let us suppose that every point of $K$ is accumulated on by a sequence of (pairwise different) circle components. Such an example is easy to c …

I think your assumption should include that the set contains at least two points, otherwise there is a trivial example ... Eric already mentioned the pseudo-arc, which is of course a perfect answer. …

I realise this is quite an old question, but here is an alternative to the already existing elegant answers, which shows that Hausdorff can be replaced by $T_1$: Proposition. No countable $T_1$ sp …