# How to define topology in terms of subobjects?

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 ).

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It seems like a better idea to internalize the notion of locale (ncatlab.org/nlab/show/locale); IIRC the notion of topological space is known to be poorly behaved in the absence of choice (so in various topoi). – Qiaochu Yuan May 3 '13 at 19:49
There's an obvious way of categorifying the third axiom: say that the subcategory has all colimits. But this is still not enough. You should ask that finite meets distribute over all joins. In any case, one will not get close to the idea of a Grothendieck topology in this way... they are not really related to topologies at all. – Zhen Lin May 3 '13 at 20:06