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$\mathbb{R}^2 \setminus C$ is path connected if $C$ is countable (Theorem)
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' \in \mathbb{R}^2\setminus C$ we may construct a path from $ d$ to $ d'$ by first following $ p_d$ and then following $ p_{d'}$ 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$. $ \qedsymbol$

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
$\displaystyle 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. $ \qedsymbol$



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Cross-references: Hausdorff spaces, connected, disjoint, open, disjoint union, uncountable, onto, continuous, completes, centre, radius, circle, distance, contain, meet, lines, straight, strategy, point, fix, path connected, subset, countable

This is version 2 of $\mathbb{R}^2 \setminus C$ is path connected if $C$ is countable, born on 2006-08-09, modified 2006-08-23.
Object id is 8234, canonical name is MathbbR2SetminusCIsPathConnectedIfCIsCountable.
Accessed 774 times total.

Classification:
AMS MSC54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

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