Irreducible topological space

hyperconnected

A topological space that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty open subset is everywhere dense. The image of an irreducible topological space under a continuous mapping is irreducible. A product of irreducible topological spaces is irreducible. The concept of an irreducible topological space is of interest only for non-separable spaces (cf. Separable space); it is constantly used in algebraic geometry, which is concerned with the non-separable Zariski topology.

An irreducible component of a topological space $X$ is any maximal irreducible subset of $X$. The irreducible components are closed and their union is the whole of $X$.

Comments

In the theory of coverings (cf. Covering (of a set)) one also has a notion of irreducibility: A topological space is irreducible if every open covering of it has an irreducible open refinement, where a covering is irreducible if no proper subfamily of it is covering. Countably-compact spaces are characterized by the condition that every irreducible open covering is finite. Thus, a space is compact if and only if it is countably compact and irreducible.

References

[a1]	R. Arens, J. Dugundji, "Remark on the concept of compactness" Portugaliae Math. , 9 (1950) pp. 141–143 MR0038642 Zbl 0039.18602