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Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$ The later relation means that $Z$ is homeomorphic …

Can anyone tell me what is the essential difference between planar sub-continua and sub-continua of the torus? I will appreciate if you can give me some references. …

It is known that a compact "indecomposable continuum" has uncountably many proper infinite subsets that are themselves "continua". … If C is a non-compact "indecomposable continuum" and S is the set of all its proper infinite subsets that are themselves "continua", what can be said about the cardinal number of S? …

What important results hold for non-metric continua, or where can I find a survey of such results? …

Arc-like continua are also called "snake-like" or "chainable" continua. For more background, see Nadler's excellent textbook 'Continuum Theory: An Introduction'. … I am writing a paper that involves arc-like continua, and I would be interested to know: Are there other interesting examples of arc-like continua that lend themselves to making nice and illuminating computer …

Using that we can deduce that for nondegenerate Peano continua $X, Y$, the spaces $C(X, I)$ and $C(I, Y)$ are quite rich and consequently the space $C(X,Y)$ (considered with the uniform topology) is quite … Questions: Are the following conditions true for all nondegenerate Peano continua $X, Y$? The point $id\in C(X,X)$ is not isolated. The space $C(X,Y)$ does not contain an isolated point. …

Define a family $\{C_i\}_{i\in I}$ of continua, that is compact connected metrizable spaces, to be hereditarily unembeddable (a name I just made up) iff for all $i\neq j$ no nontrivial subcontinuum of … Is there an uncountable family of continua which is hereditarily unembeddable? Is there such a family made of planar continua? …

The snake-like continua $\ S\ $ are universal images for Hausdorff compact spaces in the following sense: THEOREM   Let $\ S\ $ be an arbitrary snake-like continuum. … The questions is: are the snake-like continua the only universal images for the Hausdorff connected spaces? …

I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know whether another property called mixing also implies positive entropy. …

In the context of continua (i.e., non-empty compact connected metric spaces) in particular, it's well known that $X^\mathbb{N}$ is homeomorphic to the Hilbert cube, $[0,1]^\mathbb{N}$, for any dendrite … For which continua $X$ does $X^\mathbb{N}$ have the fixed-point property? …

What are examples of continua that are contractible but nowhere locally connected, meaning that no point has a neighbourhood basis consisting of connected sets? …

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point local …

I have no reason to believe that my proof for the graph-of-a-function case would easily generalize for arbitrary plane continua, so I don't know how difficult might the last question be. … It is well known that path-connected is equivalent to arc-connected (for plane continua), that is there is an arc $\gamma:[0,1]\to C$ with $\gamma(0)=D$ and $\gamma(1)=E$. …

The only result of this type I know relates to continua. Since the product of two arbitrary continua is always aposyndetic, we can guarantee that any non-aposyndetic continuum does not have a factor. …

It is a standard result that continua are $\aleph_0$-connected and, by writing a continuum as a union of singletons, it is clear that continua are $\mathfrak c$-disconnected. … Is it a theorem of $\mathsf{ZFC}$ that if $X$ and $Y$ are nontrivial continua, then $\mathrm{disc}(X)=\mathrm{disc}(Y)$? …