open and closed intervals have the same cardinality


The sets of real numbers [0,1], [0,1), (0,1], and (0,1) all have the same cardinality.

We give two proofs of this propositionPlanetmathPlanetmath.


Define a map f:[0,1][0,1] by f(x)=(x+1)/3. The map f is strictly increasing, hence injectivePlanetmathPlanetmath. Moreover, the image of f is contained in the intervalMathworldPlanetmathPlanetmath [13,23](0,1), so the maps fr:[0,1][0,1) and fo:[0,1](0,1) obtained from f by restricting the codomain are both injective. Since the inclusions into [0,1] are also injective, the Cantor-Schröder-Bernstein theorem ( can be used to construct bijectionsMathworldPlanetmath hr:[0,1][0,1) and ho:[0,1](0,1). Finally, the map r:(0,1][0,1) defined by r(x)=1-x is a bijection.

Since having the same cardinality is an equivalence relationMathworldPlanetmath, all four intervals have the same cardinality. ∎


Since [0,1] is countableMathworldPlanetmath, there is a bijection a:[0,1]. We may select a so that a(0)=0 and a(1)=1. The map f:[0,1](0,1) defined by f(x)=a(a-1(x)+2) is a bijection because it is a composition of bijections. A bijection h:[0,1](0,1) can be constructed by gluing the map f to the identity mapMathworldPlanetmath on (0,1). The formulaMathworldPlanetmathPlanetmath for h is


The other bijections can be constructed similarly. ∎

The reasoning above can be extended to show that any two arbitrary intervals in have the same cardinality.

Title open and closed intervals have the same cardinality
Canonical name OpenAndClosedIntervalsHaveTheSameCardinality
Date of creation 2013-03-22 15:43:32
Last modified on 2013-03-22 15:43:32
Owner mps (409)
Last modified by mps (409)
Numerical id 8
Author mps (409)
Entry type Result
Classification msc 26A03
Classification msc 03E10