Let's define a set (collection) [itex]\mathcal{C}[/itex] by the following conditions.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]X\in\mathcal{C}[/itex] iff all following conditions hold:

1: [itex]X\subset [0,1][/itex].

2: [itex]X[/itex] is closed.

3: If [itex]x\in X[/itex] and [itex]x<1[/itex], then there exists [itex]x'\in X[/itex] such that [itex]x<x'[/itex].

4: For all [itex]x\in X[/itex] there exists a [itex]\delta_x >0[/itex] such that [itex]]x,x+\delta_x[\;\cap\; X=\emptyset[/itex]. (So [itex]x[/itex] is not "cluster point from right".)

For example, if

[tex]

X=\big\{1-\frac{1}{n}\;\big|\; n\in\{1,2,3,\ldots\}\big\}\;\cup\;\{1\}

[/tex]

then [itex]X\in\mathcal{C}[/itex].

It is easy to construct all kinds of members of [itex]\mathcal{C}[/itex], but they all seem to be countable. My question is that does there exist an uncountable member in [itex]\mathcal{C}[/itex]?

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# Collection of continuum subsets

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