5 added a reference to Nagata's theorem

Carnahan's suggestion is the natural thing to do in the category of topological spaces, but it's unclear if we may execute it in the category of algebraic spaces, since it's unclear if the projections from the closure of the diagonal down to $X$ are always etale. Note that even for topological spaces, quotienting $X\times X$ out by the closure of the diagonal—i.e. quotienting out $X$ by the relation "$x\sim y$ if there is no pair of open neighbourhoods U of $x$ and $V$ of $y$ such that $U$ and $V$ are disjoint"—doesn't necessarily give a Hausdorff topological space since it's not an equivalence relation: it's not transitive!

But this is a technical problem: the real reason why there shouldn't exist a separification is that separatedness is a global geometric property and it's difficult to replace a scheme with another scheme for which a global geometric property holds, e.g. it's difficult to construct compactifications of schemes, even though that's relatively simple in the category of topological spaces.

At any rate, one way to construct produce a separification of a scheme is to construct produce a compactification, e.g. via Nagata's theorem.

4 added proper em-dashes

Carnahan's suggestion is the natural thing to do in the category of topological spaces, but it's unclear if we may execute it in the category of algebraic spaces, since it's unclear if the projections from the closure of the diagonal down to $X$ are always etale. Note that even for topological spaces, quotienting $X\times X$ out by the closure of the diagonal---i.ediagonal—i.e. quotienting out $X$ by the relation "$x\sim y$ if there is no pair of open neighbourhoods U of $x$ and $V$ of $y$ such that $U$ and $V$ are disjoint"---doesn't disjoint"—doesn't necessarily give a Hausdorff topological space since it's not an equivalence relation: it's not transitive!

But this is a technical problem: the real reason why there shouldn't exist a separification is that separatedness is a global geometric property and it's difficult to replace a scheme with another scheme for which a global geometric property holds, e.g. it's difficult to construct compactifications of schemes, even though that's relatively simple in the category of topological spaces.

At any rate, one way to construct a separification of a scheme is to construct a compactification.

3 replaced original text with latex, removed "EDIT"s.; deleted 2 characters in body; added 4 characters in body

Carnahan's suggestion is the natural thing to do in the category of topological spaces, but it's unclear if we may execute it in the category of algebraic spaces, since it's unclear if the projections from the closure of the diagonal down to X $X$ are always etale. (EDIT) Note that even for topological spaces, quotienting $X\times X X$ out by the closure of the diagonal---i.e., diagonal---i.e. quotienting out X $X$ by the relation "x~y $x\sim y$ if there is no pair of open neighbourhoods U of x $x$ and V $V$ of y $y$ such that U $U$ and V $V$ are disjoint"---doesn't necessarily give a Hausdorff topological space since it's not an equivalence relation: it's not transitive!

But this is a technical problem: the real reason why there shouldn't exist a separification is that separatedness is a global (EDIT) geometric property and it's difficult to replace a scheme with another scheme for which a global geometric property holds, e.g., e.g. it's difficult to construct compactifications of schemes, even though that's relatively simple in the category of topological spaces.

At any rate, one way to construct a separification of a scheme is to construct a compactification.

2 changed the original post so that it holds under Geraschenko's very correct observations.
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