the real numbers are indecomposable as a topological space

Let be the set of real numbers with standard topology. We wish to show that if is homeomorphicMathworldPlanetmath to X×Y for some topological spacesMathworldPlanetmath X and Y, then either X is one point space or Y is one point space. First let us prove a lemma:

Lemma. Let X and Y be path connected topological spaces such that cardinality of both X and Y is at least 2. Then for any point (x0,y0)X×Y the space X×Y{(x0,y0)} with subspace topology is path connected.

Proof. Let xX and yY such that xx0 and yy0 (we assumed that such points exist). It is sufficient to show that for any point (x1,y1) from X×Y{(x0,y0)} there exists a continous map σ:IX×Y such that σ(0)=(x1,y1), σ(1)=(x,y) and (x0,y0)σ(I).

Let (x1,y1)X×Y{(x0,y0)}. Therefore either x1x0 or y1y0. Assume that y1y0 (the other case is analogous). Choose paths σ:IX from x1 to x and τ:IY from y1 to y. Then we have induced paths:


Then the path σ*τ:IX×Y defined by the formulaMathworldPlanetmathPlanetmath

(σ*τ)(t)={σ(2t)when  0t12τ(2t-1)when12t1

is a desired path.

PropositionPlanetmathPlanetmath. If there exist topological spaces X and Y such that is homeomorphic to X×Y, then either X has exactly one point or Y has exactly one point.

Proof. Assume that neither X nor Y has exactly one point. Now X×Y is path connected since it is homeomorphic to , so it is well known that both X and Y have to be path connected (please see this entry ( for more details). Therefore for any point (x,y)X×Y the space X×Y{(x,y)} is also path connected (due to lemma), but there exists a real number r such that X×Y{(x,y)} is homeomorphic to {r}. ContradictionMathworldPlanetmathPlanetmath, since {r} is not path connected.

Title the real numbers are indecomposable as a topological space
Canonical name TheRealNumbersAreIndecomposableAsATopologicalSpace
Date of creation 2013-03-22 18:30:59
Last modified on 2013-03-22 18:30:59
Owner joking (16130)
Last modified by joking (16130)
Numerical id 12
Author joking (16130)
Entry type Theorem
Classification msc 54F99