An example of a space $X$ which doesn't embed in $\mathbb{R}^n$ for any $n$? Apologies if this has been asked before, but couldn't find it.
The definition of embedding that I'm using is this:

Suppose $X$ and $Y$ are topological spaces. We call a function
  $f:X\rightarrow Y$ an embedding if $f$ is a homeomorphism from $X$
  to $f(X)$, equipped with the subspace topology.

I think the idea is to look for a space $X$ where any function $f:X\rightarrow\mathbb{R}^n$ does not have a continuous inverse. I can't seem to get anywhere.

consider $X = \mathbb{R}$ with discrete metric space...then $X$ cannot be embedded in $\mathbb{R^n}$ for all $n$...since $f(X)$ would be an discrete set of $\mathbb{R^n}$..but any discrete set can atmost be countable in $\mathbb{R^n}$.
