Waste less time on Facebook — follow Brilliant.
×

Countable and (un)countable

Find an example of a set \(S\) such that \(S\) is countable, yet \(S'\) (the set of all limit points of \(S\)) is uncountable.

Note by Austin Stromme
3 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

The rationals. More generally, if \(A\) is an uncountable compact subset of a metric space then, for any \(n \in \mathbb{N}\) there is a finite subset \(S_n\) of \(A\) such that every element of \(A\) is within \(n^{-1}\) of some element of \(S_n\). Then \[S=\bigcup_{n\in\mathbb{N}}S_n\] Is a countable subset of \(A\) whose closure is \(A\).

Mark Hennings - 3 years, 5 months ago

Log in to reply

One example is the set \(S:= \{ x \colon x\in \mathbb{R}^2 \land \| x\| <1 \}\cap \mathbb{Q}^2 \).

Austin Stromme - 3 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...