On topologies defined by irreducible sets

• We introduce a new kind of derived topology a lá Scott topology.
• We systematically replace directed sets with irreducible sets.
• Results which generalize existing ones are proven, with applications to posets.
• SI-compactness is invented and related results are also obtained.

In this paper, we define and study a new topology constructed from any given topology on a set, using irreducible sets. The manner in which this derived topology is obtained is inspired by how the Scott topology on a poset is constructed from its Alexandroff topology. This derived topology leads us to a weak notion of sobriety called k-bounded sobriety. We investigate the properties of this derived topology and k-bounded sober spaces. A by-product of our theory is a novel type of compactness, which involves crucially the Scott irreducible families of open sets. Some related applications on posets are also given.