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

###### Theorem 1.

Let $C$ be a countable^{} subset of ${\mathrm{R}}^{\mathrm{2}}$. Then ${\mathrm{R}}^{\mathrm{2}}\mathrm{\setminus}C$ 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.

###### Corollary 1.

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

###### Proof.

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

$${f}^{-1}(0)\cup {f}^{-1}((-\mathrm{\infty},0))\cup {f}^{-1}((0,\mathrm{\infty}))$$ |

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.
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Title | ${\mathbb{R}}^{2}\setminus C$ is path connected if $C$ is countable |
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Canonical name | mathbbR2setminusCIsPathConnectedIfCIsCountable |

Date of creation | 2013-03-22 16:09:11 |

Last modified on | 2013-03-22 16:09:11 |

Owner | silverfish (6603) |

Last modified by | silverfish (6603) |

Numerical id | 5 |

Author | silverfish (6603) |

Entry type | Theorem |

Classification | msc 54D05 |