# How do you show the Cantor set is uncountable?

## How do you show the Cantor set is uncountable?

The canonical proof that the Cantor set is uncountable does not use Cantor’s diagonal argument directly. It uses the fact that there exists a bijection with an uncountable set (usually the interval [0,1]).

## Is the Cantor set countable?

The Cantor set is uncountable.

What points are in the Cantor set?

The Cantor set is the set of all numbers between 0 and 1 that can be written in base 3 using only the digits 0 and 2. For example, 0 is certainly in the Cantor set, as is 1, which can be written 0.2222222….

### How do you make a Cantor set?

Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments.

### How do you solve the Cantor set?

Is Cantor set totally disconnected?

A space is totally disconnected if its only connected subspaces are one-point sets. Theorem 0.3. The Cantor set is totally disconnected, and it does not have the discrete topology.

## How many points are left in the Cantor set?

If we could continue the “middle third removal” forever, it seems that eventually we would take out the entire length of the original interval from 0 to 1. However, in the final set, the Cantor Set, there are infinitely many points left, such as 0, \frac13, 1, and even weird ones like \frac14.

## Is Cantor set bounded?

Theorem: Cantor’s set is bounded. That’s because it lives inside the interval [0,1]. Theorem: Cantor’s set is closed.

Does the Cantor set contain any intervals?

The Cantor ternary set, and all general Cantor sets, have uncountably many elements, contain no intervals, and are compact, perfect, and nowhere dense.