# bijection between closed and open interval

For mapping the end points of the closed unit interval  $[0,\,1]$  and its inner points bijectively onto the corresponding open unit interval  $(0,\,1)$,  one has to discern suitable denumerable subsets in both sets:

 $\displaystyle[0,\,1]\,\;=\;\{0,\,1,\,1/2,\,1/3,\,1/4,\,\ldots\}\cup S,$ $\displaystyle(0,\,1)\;=\;\{1/2,\,1/3,\,1/4,\,\ldots\}\cup S,$

where

 $S\;:=\;[0,\,1]\smallsetminus\{0,\,1,\,1/2,\,1/3,\,1/4,\,\ldots\}.$

Then the mapping $f$ from  $[0,\,1]$  to  $(0,\,1)$  defined by

 $f(x)\;:=\;\begin{cases}1/2\quad\mbox{for}\quad x=0,\\ 1/(n\!+\!2)\quad\mbox{for}\quad x=1/n\quad(n=1,\,2,\,3,\,\ldots),\\ x\qquad\mbox{for}\quad x\in S\end{cases}$

is apparently a bijection.  This means the equicardinality of both intervals.

Note that the bijection is neither monotonic (e.g. $0\mapsto\frac{1}{2}$,  $\frac{1}{2}\mapsto\frac{1}{4}$,  $1\mapsto\frac{1}{3}$) nor continuous.  Generally, there does not exist any continuous surjective mapping  $[0,\,1]\to(0,\,1)$,  since by the intermediate value theorem a continuous function maps a closed interval to a closed interval.

## References

• 1 S. Lipschutz: .  Schaum Publishing Co., New York (1964).
Title bijection between closed and open interval BijectionBetweenClosedAndOpenInterval 2013-03-22 19:36:06 2013-03-22 19:36:06 pahio (2872) pahio (2872) 10 pahio (2872) Example msc 54C30 msc 26A30 BijectionBetweenUnitIntervalAndUnitSquare