Relations between several topologies obtained by the ideal of countable sets

The main goal of this paper is to investigate relations between topologies obtained by: $\theta$-open sets, $\omega$-open sets, ${\theta }_{\omega }$-open sets, local function, and local closure function with ideal of the countable sets. As the result we will obtain some new the relations which hold in general, and, as the ultimate goal, equality of the topology of $\omega$-open sets and topology obtained by the local function with ideal of countable sets.