-2
$\begingroup$

What is an example of a connected Hausdorff space $X$ with $|X|>1$ and a surjective continous map $f:X\to (X\times X)$?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
4
$\begingroup$

$X=[0,1]$. See Space-filling curve.

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ You can also do this with any positive dimensional connected manifold I think. $\endgroup$
    – Thomas Rot
    Commented Oct 21, 2018 at 8:17
5
$\begingroup$

The Hilbert cube $[0,1]^\omega$. Just split the coordinates in two disjoint infinite sets. Most standard sequence spaces like $\ell^\infty, \ell^2$ will similarly work.

Improve this answer
$\endgroup$
1
  • 3
    $\begingroup$ Those examples even have $X$ and $X\times X$ homeomorphic. $\endgroup$
    – Wojowu
    Commented Oct 21, 2018 at 8:47

Not the answer you're looking for? Browse other questions tagged .