In his well-known [paper](http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101110.pdf) Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:


We have an inverse-system $\{X(\alpha); f^\alpha\_\beta: \beta,\alpha < \omega\_1\}$ of metric indecomposable continua and retractions. For each $X(\alpha)$ a composant $C(\alpha) \subset X(\alpha)$ is specified and each $C(\alpha)$ maps into $C(\beta) $. 


The inverse limit $X$ has exactly two composants. The first is the union
$\bigcup\{X(\beta): \beta < \omega\_1\}$ where we identify $X(\beta)$ with the set of sequences $(x\_\alpha) \in X$ with $x\_\beta = x\_{\beta+1} = x\_{\beta+2} = \ldots . $ The second composant is the inverse limit $\{C(\alpha); f^\alpha\_\beta: \beta,\alpha < \omega\_1\}$.
Observe there is no reason *a priori* for the second composant to be nonempty. However I do not believe an example is know.


My question is an easier one. Can you think of an example of a metric continuum $M$ and a $\omega\_1$-indexed decreasing nest of dense semicontinua **with empty intersection**? We call the set $S \subset M$ a *semicontinuum* to mean for each $x,y \in S$ there exists a continuum $K$ with $\{x,y\} \subset K \subset S$.


If the second composant was empty the family $\{f^\alpha\_0(C(\alpha)): \alpha < \omega\_1\}$ would be such a nest for $M = X\_0$.


If we index by $\omega$ instead an example is easy to come by. Let $M$ be the unit disc and $Q = \{q\_1,q\_2, \ldots\}$ and enumeration of the rational points on the boundary. Let $S(n)$ be formed by drawing the straight line segment from each element of $\{q\_n,q\_{n+1}, \ldots\}$ to each rational point of $(0,1/n) \times \{0\}$. Then add in $(0,1/n) \times \{0\}$ itself to make the space a semicontinuum.


Indexing by $\omega\_1$ must somehow get around the fact that any $\omega\_1$ decreasing nest of closed subsets of a metric continuum eventually stabilizes.


It feels like this would be easier if we assume the Continuum Hypothesis.



Example of an $\omega_1$ decreasing chain of dense semicontinua?

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> I am **not** assuming any CH.
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Let $\ \Omega\ := \{\alpha: \alpha<\omega\_1\},\ $ and $\ i:\Omega\rightarrow [0;1]\ $ be injective and such that
$\ \Gamma\ :=\ i(\Omega)\ $ is condensed in $\ [0;1].\ $ More generally,
let $\ \Gamma\_\alpha\ :=\ i([0;\alpha)).\ $ Also let $\ M:=[0;1]^2.\ $
Then


$$ S\_\alpha\,\ :=\,\ (\Gamma\setminus\Gamma\_\alpha)\times[0;1]
\ \cup\ [0;1]\times(\Gamma\setminus\Gamma\_\alpha) $$


for $\ \alpha<\omega\_1,\ $ is the required decreasing $\omega\_1$-sequence which has empty intersection.




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> 
> ***Condensed*** means that every non-empty open subset of $\ [0;1]\ $ has an uncountable intersection with $\ \Gamma$.
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