A subset U C E is called open if. for every x e U, there is an open interval (a, b) where x e (a.b)
C U. Show that, in the above definition, the numbers a, b may be taken as rational; that is. if x e
U. there is an open interval (c, d) where x (c, d) C U and where c, d e Q. Show that any open set
U is a union of (possibly infinitely many) intervals (a. b) where a, b e Q. How many open
subsets of E exist?
Solution
suppose x is any element of U and is any smallestpositive real number.
then if ( x - , x + ) is in U , then U isan open set in R.
if x is a rational number , then while is a rationalnumber, we have sum and difference of rational
numbers isrational.
so, x - = a and x + = b such that x always liesbetween x - and x + which can also be written as
a< x < b or , x is in ( a,b) .
i.e. every element x of U has its neighbourhood inU.
so, U is an open set.
(b) if U is an open set , then for every x in U , we canchoose such that x is in ( x - , x +).
further, is an arbitrarily smallpositive real number and can take infinitely many values , wecan
see that there can be infinite intervals in which x lies.
further, the union of all those intervals lie inU.
thus, union of any number of open intervals is in U.
(c0 the set of real numbers is an open set and can be writtenas the union of infinite number of
open intervals..
A subset U C E is called open if. for every x e U, there is an open i.pdf
1. A subset U C E is called open if. for every x e U, there is an open interval (a, b) where x e (a.b)
C U. Show that, in the above definition, the numbers a, b may be taken as rational; that is. if x e
U. there is an open interval (c, d) where x (c, d) C U and where c, d e Q. Show that any open set
U is a union of (possibly infinitely many) intervals (a. b) where a, b e Q. How many open
subsets of E exist?
Solution
suppose x is any element of U and is any smallestpositive real number.
then if ( x - , x + ) is in U , then U isan open set in R.
if x is a rational number , then while is a rationalnumber, we have sum and difference of rational
numbers isrational.
so, x - = a and x + = b such that x always liesbetween x - and x + which can also be written as
a< x < b or , x is in ( a,b) .
i.e. every element x of U has its neighbourhood inU.
so, U is an open set.
(b) if U is an open set , then for every x in U , we canchoose such that x is in ( x - , x +).
further, is an arbitrarily smallpositive real number and can take infinitely many values , wecan
see that there can be infinite intervals in which x lies.
further, the union of all those intervals lie inU.
thus, union of any number of open intervals is in U.
(c0 the set of real numbers is an open set and can be writtenas the union of infinite number of
open intervals.
