A topological space that is neither discrete nor indiscrete

[DotKite]

Homework Statement

is it possible to have a topological space that is neither the indiscrete nor the discrete, and very set in the topology is clopen?

Homework Equations

The Attempt at a Solution

let $X$ = {(0,1),(2,3)} with the ordinary topology on R.
(0,1) is open, but it's complement which is (2,3) is open and which means (0,1) is closed. This (0,1) is clopen. Same argument for (2,3). Is this right?

 

[Dick]

Homework Statement

is it possible to have a topological space that is neither the indiscrete nor the discrete, and very set in the topology is clopen?

Homework Equations

The Attempt at a Solution

let $X$ = {(0,1),(2,3)} with the ordinary topology on R.
(0,1) is open, but it's complement which is (2,3) is open and which means (0,1) is closed. This (0,1) is clopen. Same argument for (2,3). Is this right?

Yes, it is. But you aren't stating it very well. Your space is $X$={0,1,2,3}. Your open sets are ∅,{0,1},{2,3},X. Then every open set is also closed, as you say. And that is NOT the ordinary topology on R.