Diagonal map (${\delta }_X:X\to X\times X$)

The diagonal map sends $x$ to $(x,x)$. It is an embedding of a topological space onto its diagonal subset, because either coordinate projection is its inverse on the image. For a closed oriented manifold, that subset has a fundamental class, whose Poincare dual is the cohomology class of the diagonal.

The image of the diagonal map in $X\times X$. For a manifold, the diagonal subset is an embedded submanifold diffeomorphic to $X$.