Suppose that T is a topology on R that contains the set of a

Suppose that T is a topology on R that contains the set of all closed intervals. Prove that T is the discrete topology on R.

Solution

Given:

T is a topology on R that contains the set of all closed intervals

Then for each xR,

[x,x]={x} is a closed interval. here definition of interval does not allow such degenerate intervals,

then note that [x1,x][x,x+1]={x}. Thus, {x} is open for each xR.

From above way we say T is a discrete topology on R.