Greets
I want to prove the following:
For any set $X$ there exists a Hausdorff space $Y$ and a family $\{Y_x:x\in{X}\}$ of disjoint subsets of $Y$, each dense in $Y$.
I want to prove this assertion since by proving it, I can use it to prove that any topological space is the open continuous image of a Hausdorff space.
I have proven the assertion, but using mathematical logic; namely, using the compactness theorem and the fact that the assertion is true for finite $X$ in some ordered set, but I don't know how else to prove this without the use of the compactness theorem.
I would thank very much to see a prove of this, without using mathematical logic.