How to define topology for general categories. The following is my attempt to do so but I guess it is not correct. What should be the categorical analouge of the third axiom of topology (that the union of open sets is open.)
A topology for an object $X \in C$ is a subcategory $\Omega$ of the category of subobjects of $X$, which:
contains the terminal object of $Sub\left(X\right)$, namely $id: X\rightarrow X $ and the empty suobject (the no morphism from the initial object of $C$ to X). This condition is generalised from of saying that $X$ and the empty set are open.
$\Omega$ is closed under finite pullbacks.(The pullback of any finite number of monics with codomain $X$ is included in $\Omega$). This must resemble the closure under intersection.
Finally, I wanted to know this attempt is any close to the idea of sieve (or of Grothendiek's Topology ).