Density of [0,1]-C on Thick Cantor Set Construction | Sequence {a_n}

In summary, the construction of the thick Cantor set involves removing the middle a_n part of each interval at the n-th stage. The goal is to show that [0,1]-C (where C is the thick Cantor set) is dense. One way to prove this is by showing that C is nowhere dense. This can be done by identifying points of Cantor's set with sequences of 0 and 1 and finding a sequence in [0,1]-C that converges to an arbitrary point in C.
  • #1
hj2000
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0
Let C be the thick Cantor set. let {a_n} be a sequence of positive numbers.
In the construction of the thick Cantor set, at the n-th stage we remove the middle a_n part of each interval (instead of the middle third as in the ordinary Cantor set).

I actually wanted to show that [0,1]-C is dense (where C is the thick Cantor set). How do I show it?
 
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  • #3
I would prove the following. Let [itex]x\in C[/itex] be arbitrary. It suffices to find a sequence [itex]x_n\in [0,1]-C[/itex] so that [itex]x_n\to x[/itex]. Do you know how to identify points of Cantor's set with sequences of 0 and 1? I mean sequences like 0011... 1010... 1101... and so on.
 

1. What is a Fat Cantor Set?

A Fat Cantor Set is a mathematical set that is formed by removing the middle third of a line segment, then removing the middle third of each remaining segment, and repeating this process infinitely. This creates a set of points with zero length, but with a positive measure.

2. How is the Fat Cantor Set different from the Cantor Set?

The Fat Cantor Set is a variation of the Cantor Set, which is formed by removing the middle third of a line segment repeatedly until there are no segments left. The difference is that the Fat Cantor Set leaves some segments behind, resulting in a set with positive measure instead of zero measure.

3. What is the importance of the Fat Cantor Set in mathematics?

The Fat Cantor Set is an example of a fractal, which is a geometric shape that exhibits self-similarity at different scales. It has important applications in chaos theory, dynamical systems, and measure theory.

4. Can the Fat Cantor Set be visualized?

Yes, the Fat Cantor Set can be visualized by plotting the points that remain after each iteration of removing the middle third. It will appear as a jagged line with gaps in between, similar to the Cantor Set but with some segments left behind.

5. How does the Fat Cantor Set relate to real-life phenomena?

The Fat Cantor Set can be used to model real-life phenomena that exhibit self-similarity, such as the branching patterns of trees, river networks, and lightning bolts. It also has applications in computer graphics and image compression.

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