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

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

is countable

(Theorem)

Theorem 1 Let 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

not in

. The strategy of the proof is to construct a path

from any $x \in \mathbb{R}^2\setminus C$ to

. If we can do this then for any $d, d' \in \mathbb{R}^2\setminus C$ we may construct a path from

by first following

and then following $p_{d'}$ in reverse.

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

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

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

is continuous), non-empty (as

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 MSC:

54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

Pending Errata and Addenda

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