ag.algebraic geometry - Schemy question..

Here is where I got lost .. I have a scheme Y over k (an algebraically
closed field), in it I have an irreducible closed subscheme X of
finite type (do I need finite type?). I also know that X is
universally closed (over k) and separated (do I need separated?) ..
then why is it that X should be either a point or Y itself?

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