Let $X$ be a compact topological space and let $\{A_\alpha\}_{\alpha\in \Delta}$ be a family of non-compact topological subspaces of $X$ such that $\cup_\alpha A_\alpha=X$ and $A_\alpha\cong A_\beta$ for all $\alpha,\beta\in \Delta$.
Is there an induced topological space $Y$ such that the correspondent subspaces $\{A'_\alpha\}_{\alpha\in \Delta}$ are compact in $Y$. In other words, is there anything one can do "like compactification" to $X$ or the subspaces to get compact subspaces?