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In this chapter, two additional properties of closure operators are introduced. Their relationship with idempotency, weak heredity and infima and suprema of closure operators is discussed.

We have already seen in the previous chapter that although always idempotent, regular closure operators need not be weakly hereditary or additive. In this chapter we provide some sufficient conditions for a regular closure operator to be hereditary. Some conditions that imply and are equivalent to weak heredity of a regular closure operator will be presented in the next chapter after the relationship between regular closure operators and epimorphisms has been cleared up.

In this chapter we show that by adding some mild assumptions that revolve around the existence of a terminal object, most of the classical properties of topological connectedness hold in our more general setting. Moreover, some interesting characterizations of the notions of (C, N-connected and (C, N)-disconnected objects introduced in the previous chapter, can be given.

In this survey article, we indicate how some of the recent ideas and techniques introduced in the study of infinite rank Butler groups can be successfully used in the investigation of the homological dimensions of torsion-free modules over integral domains and, in particular, over valuation domains.

A right noetherian ring R that has local bijective Gabriel correspondence with respect to a torsion theory τ need in general not be right fully τ-bounded, but it is, if and only if the τ -closed prime ideals satisfy a version of the second layer condition. Rings with local bijective Gabriel correspondence are characterized by the tameness of their τ -torsionfree modules, and this implies that their relative (Gabriel-Rentschler) Krull and classical Krull dimensions coincide.