Skip to main content
added 1 character in body; edited title
Source Link
Anton Petrunin
  • 45k
  • 14
  • 135
  • 299

Can a connected planar compactum minus a point be totally disconnected?  

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$$X\smallsetminus\{x\}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

Can a connected planar compactum minus a point be totally disconnected?  

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

Can a connected planar compactum minus a point be totally disconnected?

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x\}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

Added backticks
Source Link
HJRW
  • 25k
  • 3
  • 68
  • 144
Added condition
Source Link
HJRW
  • 25k
  • 3
  • 68
  • 144

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ and let $x\in X$. Can $X\smallsetminus\{x}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

What the title said. In a slightly more leisurely fashion:-

Let $X$ be a compact, connected subset of $\mathbb{R}^2$ with more than one point, and let $x\in X$. Can $X\smallsetminus\{x}$ be totally disconnected?

Note that the Knaster-Kuratowski fan shows that, in the absence of the compactness hypothesis, the answer can be 'yes'.

To give credit where it's due, this question was inspired by one that I was asked by Barry Simon.

Fixed link
Source Link
Bjorn Poonen
  • 23.8k
  • 7
  • 90
  • 109
Loading
Source Link
HJRW
  • 25k
  • 3
  • 68
  • 144
Loading