Can $\mathbb{R}$ be homeomorphic to some $X \times X$?

Question

I got this question in my topology exam and I had no idea how to make it, topology can be so weird sometimes it's hard to imagine some spaces.

My heart told me that it is not possible, but I do not know how to proceed on this. The easiest way seems to be using some topological invariant, line connectedness.

Please note that the factors on the product are the same.

wish I could +1 more than once for "my heart told me that it is not possible" instead of the overused "it seems intuitively false" — hunter, Mar 08 '15 at 22:24
I think projecting onto the first factor produces a problem. — Adam Hughes, Mar 08 '15 at 22:26
See also here, here, and the links provided at some of the answers. — Andrés E. Caicedo, Mar 08 '15 at 22:36

score 1 · Accepted Answer · answered Mar 08 '15 at 22:27

1

Hint: Think about path-connectedness of the spaces in consideration.

answered Mar 08 '15 at 22:27

Stahl

23,212

Can $\mathbb{R}$ be homeomorphic to some $X \times X$?

1 Answers1