The Banach space `∞/c0 is isomorphic to the linear space of continuous functions on N∗ with the supremum norm, C(N∗). Similarly, the canonical representation of the `∞ sum of `∞/c0 is the Banach space of continuous functions on the closure of any non-compact cozero subset of N∗. It is important to determine if there is a continuous linear lifting of this Banach space to a complemented subset of...

For Tychonoff X and α an infinite cardinal, let αdef X := the minimum number of α cozero-sets of the Čech-Stone compactification which intersect to X (generalizing R-defect), and let rtX := minα max(α, α def X). Give C(X) the compact-open topology. It is shown that τC(X) ≤ nχC(X) ≤ rtX = max(L(X), L(X) def X), where: τ is tightness; nχ is the network character; L(X) is the Lindelöf number. For ...

An archimedean vector lattice A might have the following properties: (1) the sigma property (σ): For each {an}n∈N ⊆ A there are {λn}n∈N ⊆ (0,∞) and a ∈ A with λnan ≤ a for each n; (2) order convergence and relative uniform convergence are equivalent, denoted (OC⇒ RUC): if an ↓ 0 then an → 0 r.u. The conjunction of these two is called strongly Egoroff. We consider vector lattices of the formD(X)...

We generalize a major portion of the classical theory of Cand C embedded subspaces to pointfree topology, where the corresponding notions are frame Cand C -quotients. The central results characterize these quotients and generalize Urysohn’s Extension Theorem, among others. The proofs require calculations in CL, the archimedean f -ring of frame maps from the topology of the reals into the frame ...

In this paper, we study the concept of $mathcal C$-reticulation for the category $mathcal C$ whose objects are lattice-valued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$-reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$-reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the conce...