A real x is 1 1-Kurtz random (1 1-Kurtz random) if it is in no closed null 1 1 set (1 1 set). We show that there is a cone of 1 1-Kurtz random hyperdegrees. We characterize lowness for 1 1-Kurtz randomness as being 1 1-dominated and 1 1semi-traceable.