Answer to Question #297007 in Discrete Mathematics for Joe

Question

SHOW that every open interval is uncountable

Answer 1

Prove this theorem same thing to do as above,here just set x_n=((a+b)/2).a_n1a_n2a_n3.....a_nn..... and y=((a+b)/2).y₁y₂y₃.....y_n.....

We will prove this theorem by contradiction.

Assume that A is open and A=(a,b).A is countable also.

Since,B=[a,b] is a uncountable set and we know that If B is an uncountable set and A is a countable set,then (B-A) is uncountable set.

Here,B-A=[a,b]-(a,b)={a,b},which is consists of two elements and is finite,so countable,which is contradiction.

This, If A is countable then A can not be open set.