Irreducible completion of topological spaces

Abstract

It has long been known that in a $T_0$ space, directed sets with respect to the specialization order are irreducible sets. Motivated by directed completion of topological spaces, we use irreducible sets to define SI-spaces and irreducible completion of topological spaces, called SI-completion. Then we give a SI-completion of topological spaces. This completion is idempotent. And we find SI-spaces are exactly sober spaces and show that the category $ \mathbf{SOB^+} $ consisting of all sober spaces with SI-continuous map is a reflective subcategory of $ \mathbf{TOP_0^+} $ consisting of all $T_0$ spaces with SI-continuous map. Inspired by the fact that irreducible sets of a poset with respect to the Alexandroff topology are exactly directed sets and the space is monotone determined, we study the connections between directed completion and irreducible completion and find that $\Gamma (P(SI(L_\gamma ))) \hookrightarrow \Gamma (\Sigma (cl_d(\psi (L))))$ for a poset $L$.