I am self-studying the general topology these day and find that the third axiom of the topological space $(X,\tau)$ defined by open set is:
For any finite collection of $U_i \in \tau$, the intersection of $U_i$ is still a member of $\tau$.
Now I am wondering is there any space$(X,\tau)$ such that satisfying the following conditions:
$1)\emptyset , X \in \tau$
$2)$For any finite or infinite collection of $U_i \in \tau$, the union of $U_i$ is still a member of $\tau$.
$3)$For any finite or infinite collection of $U_i \in \tau$, the intersection of $U_i$ is still a member of $\tau$.
I come up with space $(X,2^X)$ satisfying this condition but I cannot find non-trivial one satisfying such condition.
Does such a space exists? What name is it?