Our work shows that, for any given ideal \(\mathcal {I}\) on a non empty set X, we can find a topology \(\tau \) in which the set of all closed and discrete sets in X coincides with \(\mathcal {I}\) . We prove that if \(\mathcal {I}\) is any proper ideal on X, then we can find a topology \(\tau ^{\prime }\) in which \(\tau ^{\prime }\) makes \(\mathcal {I}\) closed and discrete and \(\tau ^{\prime }\) is \(T_{0}\) . Furthermore, we derive some properties of this topology. Finally, we find a compact extension of the newly constructed space.
