Let $X$ and $Y$ be manifolds, and let $f$ be a mapping $f: X \to Y$ Choose $x \in X$ and let $y=f(x)$ Recall that $df_x: T_x(X) \to T_y(Y)$ is the derivative of $f$ at $x$ and $T_z(Z)$ is the tangent space of manifold $Z$ at point $z$

If $df_x$ is injective, then $f$ is said to be an immersion at x. If $f$ is an immersion at every point, it is called an immersion.

If the image of $f$ is also closed, then $f$ is called a closed immersion.

The notion of closed immersion for schemes is the analog of this notion in algebraic geometry.