TITLE: $\mathbb R^2$ is not homeomorphic to $\mathbb R^3$.
QUESTION [9 upvotes]: I was reading Munkres for Topology. In that, it is mentioned that $\mathbb R$ is not homeomorphic to $\mathbb R^2$ as deleting a point from both makes the first one disconnected while the latter one still remains connected.
Can't we say on the same lines that $\mathbb R^2$ is not homeomorphic to $\mathbb R^3$ as deleting a line from both makes the first one disconnected but second one is still connected.
Where I'm going wrong?
EDIT
I know we can show them to be non homeomorphic using simple connectedness, but I want to find the error in that approach.

REPLY [9 votes]: Key fact: "$x$ is a point in topological space $X$" is a property invariant under homeomorphism, whereas "$x$ is a line in topological space $X$" (whatever definition of line you take) is not invariant under homeomorphism.
The argument that $\mathbb{R}$ and $\mathbb{R}^2$ are not homeomorphic goes through because, supposing $\mathbb{R}$ were somehow homeomorphic to $\mathbb{R}^2$ we know the removed point in $\mathbb{R}$ corresponds to a point in $\mathbb{R}^2$. But not so with a line in $\mathbb{R}^2$ corresponding to a line in $\mathbb{R}^3$.
Of course, you could instead of saying remove a line from $\mathbb{R}^2$, remove a copy of $\mathbb{R}$ (a subspace homeomorphic to $\mathbb{R}$. This is now invariant under homeomorphism, however, you have essentially the same problem: a copy of $\mathbb{R}$ in $\mathbb{R}^3$ need not be anything nice, and certainly need not be a line.

Another way of saying this is that the "line" in $\mathbb{R}^3$ can't be chosen by you; it has to be the image of the line in $\mathbb{R}^2$ under the homeomorphism.
To show topological spaces $X$ and $Y$ are not homeomorphic, it certainly is not sufficient to remove the same subspace from both (that you pick) and show that the results are not homeomorphic spaces. A counterexample is deleting the open interval $(0,1)$ from $\mathbb{R}$ and $(0,2)$ (homeomorphic topological spaces); the former becomes disconnected; the latter remains connected.