Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$

Solution 1:

Here's a slightly different way of looking at it that avoids fundamental groups (although has its own messy details to check). One of the spaces, upon removing a compact set, can be separated into two connected components with noncompact closure. The other can't.

Solution 2:

Let me give a solution to the problem that uses the idea of homotopy of loops but avoids that of the fundamental group or the co/homology groups. (This is actually a fleshing out of Christian Blatter's comment to Matt's answer. FWIW I thought of this independently a couple of hours ago but had to run off to proctor an exam for someone else's class.)

We say that two loops $\gamma_0, \gamma_1: S^1 \rightarrow X$ are homotopic if there exists a continuous function $G: S^1 \times [0,1] \rightarrow X$ such that for all $x \in S^1$, $G(x,0) = \gamma_0(x)$ and $G(x,1) = \gamma_1(x)$. (Note I am not fixing a basepoint here: it doesn't matter either way for what I am about to say.)

We say that a topological space $X$ is simply connected if it is connected and every loop in $X$ is homotopic to a constant loop. It is clear that if $X$ is simply connected and $Y$ is not, then $X$ cannot be homeomorphic to $Y$.

I claim that $\mathbb{R}^2$ is simply connected and $\mathbb{R}^2 \setminus \{0\}$ is not.

Step 1: We show directly from the definition that $\mathbb{R}^2$ is simply connected.
Indeed, if $\gamma: S^1 \rightarrow \mathbb{R}^2$ is any loop, then the map $G: S^1 \times [0,1]$, $G(x,t) = (1-t)\gamma(x)$ is a homotopy from $\gamma$ to the constant loop based at $0$.

Step 2: We show that $\mathbb{R}^2 \setminus \{0\}$ is not simply connected. For this we exploit the fact that $\mathbb{R}^2 \setminus \{0\}$ is an open subset in the complex plane and use complex analysis, specifically:

Theorem (Homotopy Form of Cauchy's Integral Theorem): Let $\Omega$ be an open subset of the complex plane, let $f$ be a holomorphic function on $\Omega$ and let $\gamma_1,\gamma_2$ be two homotopic paths in $\Omega$. Then $\int_{\gamma_1} f(z) dz = \int_{\gamma_2} f(z) dz$.

This is a nontrivial result, but it is a standard variant of the usual Cauchy Integral Formula that is found in most basic complex analysis texts. In particular, if $\Omega$ is simply connected, then every loop $\gamma: S^1 \rightarrow \Omega$ is homotopic to a constant loop and thus for every holomorphic function $f$ on $\Omega$ we have $\int_{\gamma} f(z)dz = 0$. In particular this applies to $\Omega = \mathbb{C}$ by Step 1.

The endgame is probably familiar: on $\mathbb{C} \setminus \{0\}$, if you integrate the holomorphic function $f(z) = \frac{1}{z}$ on the path $\gamma(t) = e^{2 \pi i t}$ then you get $2 \pi i$, which is not zero. Thus $\mathbb{C} \setminus \{0\}$ is not simply connected and hence not homeomorphic to $\mathbb{C}$.

Solution 3:

$\mathbb{R}^2\backslash\{(0,0)\}\cong S^1\times\mathbb{R}$. you can use co/homology or fundamental groups (if that's in your toolkit). you can note that one is contractible and the other is homotopy equivalent to a circle (and tell those apart). you can note that one has euler characteristic 0 and the other has euler characteristic 1 (if this is something you can calculate).