Let $\kappa$ be a cardinal, and let $\text{Top}(\kappa)$ be the set of topological spaces $(X,\tau)$ such that $X\subseteq \kappa$. We pre-order $\text{Top}(\kappa)$ by 



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> > for $X, Y \in \text{Top}(\kappa)$ we set $X \to Y$ iff there is a continuous surjective map $f:X\to Y$.
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We say $X\simeq Y$ iff $X\to Y$ and $Y\to X$. So the relation $\geq$ on $\text{Top}(\kappa)/\simeq$ defined by $[X]\_\simeq \geq [Y]\_\simeq$ iff $X\to Y$ is well-defined, and $(\text{Top}(\kappa)/\simeq, \geq)$ is a poset.


Is $(\text{Top}(\kappa)/\simeq, \geq)$ also a lattice? Is it complete? Has it been studied anywhere?



Continuous image relation on topological spaces
A partial answer: It is not a lattice, at least if you allow non-Hausdorff spaces. 


For any natural numbers $n$ and $k$, write $\bf n+k$ for the sum of an indiscrete space of size $n$ (i.e., a space that has no nontrivial open sets) and an indiscrete space of size $k$. Then both $\bf 4$ and $\bf 2+1$ are lower bounds for the set $\{\bf 2+2, \, 3+1 \}$, but there is no greatest lower bound. 


