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: Set theory. Schaum Publishing Co., New York (1964).

Title	bijection between closed and open interval
Canonical name	BijectionBetweenClosedAndOpenInterval
Date of creation	2013-03-22 19:36:06
Last modified on	2013-03-22 19:36:06
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Example
Classification	msc 54C30
Classification	msc 26A30
Related topic	BijectionBetweenUnitIntervalAndUnitSquare