= Solution
A <quasi-projective variety> is a locally <closed subset> of <projective space>: equivalently an <open subset> of some <projective variety>, with the induced <algebraic variety> structure. <Affine varieties> are examples, via the standard <affine chart> in <projective space>.
It is <irreducible topological space>[irreducible] if it is nonempty and cannot be written as the <union> of two proper <closed subsets>. Equivalently, every two nonempty <open subsets> meet, or every nonempty <open subset> is <dense>. Closed and open are interpreted in the induced <Zariski topology> of $X$.
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