2C is path connected if C is countable

Theorem 1.

Let C be a countableMathworldPlanetmath subset of R2. Then R2C is path connected.

We use 2 simply as an example; an analogous proof will work for any n,n>1.


Fix a point P not in C. The strategy of the proof is to construct a path px from any x2C to P. If we can do this then for any d,d2C we may construct a path from d to d by first following pd and then following pd in reverse.

Fix x2C, 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 distancePlanetmathPlanetmath 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 2C 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 completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath our path form x to P. ∎

Corollary 1.

Let f:R2R be continuousPlanetmathPlanetmath and onto. Then f-1(0) is uncountable.


Suppose that f-1(0) is countable. 2 can be written as the disjoint unionMathworldPlanetmath


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 connectedPlanetmathPlanetmath for Hausdorff spaces, we have that 2f-1(0) is not path connected, contradicting the theorem. ∎

Title 2C is path connected if C is countable
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