Today I occasionally got a generalization of the concept of topology, and I call it a "skew topology" on some special categories.
Suppose $\mathcal{C}$ be a category, such that for any pair (A,B) of objects in $\mathcal{C}$, Hom(A,B) is either empty or a singleton, and that A and B are isomorphic if and only if A = B. Clearly the morphisms are analogous to a patrial order. Suppose $\mathcal{C}$ to have an minimum object up to the "order" by the morphisms, say $\phi$; and a maximal object, say $M$. Let $\wedge$ and $\vee$ are two operators on $\mathcal{C}$. If $\mathbf{T}$ is a collection (not necessary a set) of objects in $\mathcal{C}$, satisfying: (1)$\phi \in \mathbf{T}$ and $M \in \mathbf{T}$; (2)for any $A$ and $B$ in $\mathbf{T}$, $A \wedge B \in \mathbf{T}$; (3)for any family of objects $A_i$ in $\mathbf{T}$ indexed by $i \in I$, $\bigvee_{i\in I}\ A_i \in \mathbf{T}$. Then $\mathbf{T}$ is called a $\mathbf{skew}$ $\mathbf{topology}$.
In the case of point-set topology, let $\mathcal{C}$ be the category with objects be sets, morphisms be the $\subset$ relation, union of sets be $\vee$ and intersection be $\wedge$. Let $\mathbf{T}$ be the power set of the underlying set. So the concept of topology fits with the so called "skew topology" concept here. What's more, in the "skew topology" of all algebraic extensions of a field $\mathbb{F}$ with the lattice structure, the algebraic closure is just the "skew topological closure".
Could you please tell me whether there has been a concept satisfying my "skew topology" notation? If so, what is it? I'm very sorry that my English is not well, and that my expression may be not clear. I will explain if I made any confusion.