# $\mathbb{R}^{2}\setminus C$ is path connected if $C$ is countable

###### Theorem 1.

Let $C$ be a countable  subset of $\mathbb{R}^{2}$. Then $\mathbb{R}^{2}\setminus C$ is path connected.

We use $\mathbb{R}^{2}$ simply as an example; an analogous proof will work for any $\mathbb{R}^{n},n>1$.

###### Proof.

Fix a point $P$ not in $C$. The strategy of the proof is to construct a path $p_{x}$ from any $x\in\mathbb{R}^{2}\setminus C$ to $P$. If we can do this then for any $d,d^{\prime}\in\mathbb{R}^{2}\setminus C$ we may construct a path from $d$ to $d^{\prime}$ by first following $p_{d}$ and then following $p_{d^{\prime}}$ in reverse.

Fix $x\in\mathbb{R}^{2}\setminus C$, and consider the set of all (straight) lines through $x$. There are uncountably many of these and they meet in the single point $x$, so not all of them contain a point of $C$. Choose one that doesn’t and move along it: your distance  from $P$ takes on uncountably many values, and hence at some point this distance $r$ from $P$ is not shared by any point of $C$. The whole of the circle with radius $r$, centre $P$, lies in $\mathbb{R}^{2}\setminus C$ so we may move around it freely.

Consider all lines through $P$: these all intersect this circle, and there are uncountably many of them so we may choose one, say $L$, that contains no point of $C$. Moving around the circle until we meet $L$ and then following it inwards completes      our path form $x$ to $P$. ∎

###### Corollary 1.

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be continuous  and onto. Then $f^{-1}(0)$ is uncountable.

###### Proof.

Suppose that $f^{-1}(0)$ is countable. $\mathbb{R}^{2}$ can be written as the disjoint union  $f^{-1}(0)\cup f^{-1}((-\infty,0))\cup f^{-1}((0,\infty))$

where the last two sets are open (as $f$ is continuous), non-empty (as $f$ is onto) and disjoint. Since pathwise connected is the same as connected  for Hausdorff spaces, we have that $\mathbb{R}^{2}\setminus f^{-1}(0)$ is not path connected, contradicting the theorem. ∎

Title $\mathbb{R}^{2}\setminus C$ is path connected if $C$ is countable mathbbR2setminusCIsPathConnectedIfCIsCountable 2013-03-22 16:09:11 2013-03-22 16:09:11 silverfish (6603) silverfish (6603) 5 silverfish (6603) Theorem msc 54D05