  A net $(x_\alpha)$ in a Banach lattice $X$ is said to un-converge to a vector
$x$ if $\bigl\lVert\lvert x_\alpha-x\rvert\wedge u\bigr\rVert\to 0$ for every
$u\in X_+$. In this paper, we investigate un-topology, i.e., the topology that
corresponds to un-convergence. We show that un-topology agrees with the norm
topology iff $X$ has a strong unit. Un-topology is metrizable iff $X$ has a
quasi-interior point. Suppose that $X$ is order continuous, then un-topology is
locally convex iff $X$ is atomic. An order continuous Banach lattice $X$ is a
KB-space iff its closed unit ball $B_X$ is un-complete. For a Banach lattice
$X$, $B_X$ is un-compact iff $X$ is an atomic KB-space. We also study
un-compact operators and the relationship between un-convergence and
weak*-convergence.
