Open Ultrafilter Research Articles

Bovine Serum Albumin Rejection by an Open Ultrafiltration Membrane: Characterization and Modeling.

The classic application of ultrafiltration (UF) is for the complete retention of proteins, and in that situation, the transport behavior is well established. More open membranes with fractional retention are used when separating different proteins. However, protein transport has not been well documented yet in the literature. The bovine serum albumin (∼69 kDa) observed rejection ranges from 0.65 to 1 using a 300 kDa molecular weight cut-off membrane at different pH, ionic strength, and pressure. We demonstrated that, especially with open UF, the transport of proteins through the membrane is dominated by advection, with insignificant diffusion effects (p value > 0.05). We showed that with open UF, retention is not only caused by size exclusion but also to a large extent by electrostatic interactions and oligomerization of the proteins. Mass transfer in the polarization layer was relatively independent of the pH and ionic strength. It was underestimated by common Sherwood relations due to a relatively large contribution of the reduction in the flow turbulence near the membrane by the removal of fluid through the membrane. We propose a model that allows relatively quick characterization of the rejection of proteins without prior knowledge of the pore sizes and charges based on just a limited set of experiments. Therefore, protein rejection with the open UF system can be targeted by tuning the processing conditions, which might be useful for designing protein fractionation processes.

Open Ultrafilters and Separability with the Use of the Operation of Closure

We study ultrafilters of topologies as well as sets of ultrafilters that each time dominate the open neighborhood filter of some fixed point in a topological space. The sets of ultrafilters are considered as “enlarged points” of the original space. We study conditions that provide the distinguishability of (enlarged) “points” of this type. We use nontraditional separability axioms and study their connection with the known axioms T0, T1, and T2.

On the cardinality of Hausdorff spaces and H-closed spaces

We introduce the cardinal invariant aL′(X) and show that |X|≤2aL′(X)χ(X) for any Hausdorff space X (a corollary of Theorem 4.4). This invariant has the properties a) aL′(X)=ℵ0 if X is H-closed, and b) aL(X)≤aL′(X)≤aLc(X). Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound 2L(X)χ(X) from which it follows that |X|≤2ψc(X) if X is H-closed (Dow/Porter [5]). The invariant aL′(X) is constructed using convergent open ultrafilters and an operator cˆ:P(X)→P(X) with the property clA⊆cˆ(A)⊆clθ(A) for all A⊆X. As a comparison with this open ultrafilter approach, in §3 we additionally give a κ-filter variation of Hodel's proof [8] of the Dow–Porter result. Finally, for an infinite cardinal κ, in §5 we introduce κwH-closed spaces, κH′-closed spaces, and κH″-closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if κ is an infinite cardinal and X a κwH-closed space with a dense set of isolated points such that χ(X)≤κ, then |X|≤2κ, and b) if X is κH′-closed or κH″-closed then aL′(X)≤κ. This latter result relates these notions to the invariant aL′(X) and the operator cˆ.

Membrane treatment of the bleaching plant (EPO) filtrate of a kraft pulp mill

The objective of this study was to evaluate the use of membrane technology to treat oxygen and peroxide-reinforced extraction stage (EPO) filtrate from a kraft pulp mill bleach plant. Three different types of tubular membranes were tested in a pilot plant: (i) tight ultrafiltration (UF); (ii) open UF followed by nanofiltration (UF+NF); and (iii) nanofiltration (NF). According to the separation performance, considering the chemical oxygen demand (COD) and colour removal, permeate flux, operational simplicity and cost, the results indicated that the best option for treatment of (EPO) filtrates was the tight UF membrane. This membrane obtained a COD removal of 79% with a colour reduction of 86%. The effect of (EPO) filtrate UF treatment on the mill effluent treatment plant was evaluated. Compared with the actual mill effluent, the results indicated that if the UF permeate was recycled in the bleaching area, the COD reduction efficiency increased by 7%, the final effluent colour decreased by 8%, the biological sludge production decreased by 18%, and the energy consumption decreased by 40%. In the tertiary treatment plant, the coagulant dosage decreased by 40%, and the tertiary sludge production decreased by 46%.

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω).

On open ultrafilters and maximal points

In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1–45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter. Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195–207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157–172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216–266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime. Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T 1 -topologies? Appl. Gen. Topol. 5 (2004) 231–242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619–624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187–225].

Conversion of sodium lactate to lactic acid with water-splitting electrodialysis.

The conversion of sodium lactate to lactic acid with water-splitting electrodialysis was investigated. One way of reducing the power consumption is to add a conductive layer to the acid compartment. Doing this reduced the power consumption by almost 50% in a two-compartment cell, whereas the electric current efficiency was not affected at all. Three different solutions were treated in the electrodialysis unit: a model solution with 70 g/L of sodium lactate and a fermentation broth that had been prefiltered two different ways. The fermentation broth was either filtered in an open ultrafiltration membrane (cut-off of 100,000 Dalton) in order to remove the microorganisms or first filtered in the open ultrafiltration membrane and then in an ultrafiltration membrane with a cut-off of 2000 Dalton to remove most of the proteins. The concentration of sodium lactate in the fermentation broth was 70 g/L, as well. Organic molecules present in the broth (peptides and similar organic material) fouled the membranes and, therefore, increased power consumption. Power consumption increased more when permeate from the more open ultrafiltration membrane was treated in the electrodialysis unit than when permeate from the membrane with the lower cut-off was treated, since there was a higher amount of foulants in the former permeate. However, the electrodialysis membranes could be cleaned efficiently with a 0.1 M sodium hydroxide solution.

On strict and simple type extensions

Let (Y, τ) be an extension of a space (X, τ′) · p ∈ Y, let . For U ∈ τ′, let . In 1964, Banaschweski introduced the strict extension Y#, and the simple extension Y+ of X (induced by (Y, τ)) having base {o(U) : U ∈ τ′} and , respectively. The extensions Y# and Y+ have been extensively used since then. In this paper, the open filters ℒp = {W ∈ τ′ : W⫆intxclx(U) for some , and = {W ∈ τ′ : intxclx(W) ∈ ℒp} = ∩{𝒰 : 𝒰 is an open ultrafilter on on X are used to define some new topologies on Y. Some of these topologies produce nice extensions of (X, τ′). We study some interrelationships of these extensions with Y#, and Y+ respectively.

Accumulation points of nowhere dense sets in $H$-closed spaces

In this paper a ten year old problem by Kulpa and Szymanski is settled by constructing an example of a minimal Hausdorff space without isolated points which has a point that is not the accumulation point of any nowhere dense subset of the minimal Hausdorff space. Also, a result by Kulpa and Szymanski is extended by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. A decade ago, Kulpa and Szymafnski [KS] showed that each point in a compact Hausdorff space without isolated points is the accumulation point of a nowhere dense set and they asked if the result is true in the setting of minimal Hausdorff spaces without isolated points. They noted that the result is false for arbitrary H-closed spaces without isolated points. In this paper, we settle this question by giving an example of a minimal Hausdorff space without isolated points and a point which is not the accumulation point of any nowhere dense set. Also, we extend Kulpa and Szymanski's result by showing that a regular point in an H-closed space without isolated points is the accumulation point of some nowhere dense subset. First, a few preliminary results and definitions are needed. All spaces considered in this paper are assumed to be Hausdorff. A space X is H-closed if X is closed in every space in which X is a subspace. An open set U in a space X is said to be regular open if U = int(cl U). A space X is semiregular if { U c X: U is regular open} is a base for the topology of X and is minimal Hausdorff if there is no strictly coarser Hausdorff topology on the space. Katetov [K] has shown that a space is minimal Hausdorff iff it is H-closed and semiregular. For a space X, let tX = X U {q: qi is a free open ultrafilter on X}. For each open set U c X, let oU= U U { E It X\X: int(clV) c U for some Ve 9/}. THEOREM 1. Let X be a semiregular space. The following are true: (a) [B, F, S] The family {oU: U is regular open in X} is a base for a minimal Hausdorff extension ttX of X. (b) [PW] For a regular open subset U of X, cl,,XoU = oU U cl xU. D Received by the editors May 29, 1984. 1980 Mathematics Subject Classification. Primary 54D25.

Regular closed maps

A subset A A of X X is far from the remainder if whenever U \mathcal {U} is a free open ultrafilter in X X there exists U ∈ U U \in \mathcal {U} such that A ∩ c l X U = ∅ A \cap {\text {c}}{{\text {l}}_X}U = \emptyset . A map is regular closed provided that the image of every regular closed set is closed. In this note we use some recent results of G. Viglino to show that every map can be extended to a regular closed map with far from the remainder point inverses. We also relate these maps to several other interesting classes of maps.

The α-Closure αX of a Topological Space X

Introduction. It is known that every completely regular space S has a real compactification vX, contained in ,BX, with the following property: every fE C(X) has an extension fE C(vX) [2, p. 1 18 ]. (A space S is real-compact iff every Z-ultrafilter with the countable intersection property is fixed. A real compactification of X is a real compact space in which S is densely imbedded.) This paper is a study of a-spaces (every open ultrafilter with the countable intersection property converges). The main results are contained in ?III. We show that for any space X, there exists an a-closure (see Definition 3.7) aX of X, contained in KX [3, p. 89], with the following property: If Y is any other a-closure of S and i: X-* Y is the inclusion, then i can be extended continuously to a function 1: aX-* Y. Since the construction of aX is based on KX and the structure of KX is related to open ultrafilter, therefore in ?I, we study open ultrafilters and in ?II, we state a main theorem about KX. The author wishes to thank the referee for his suggestions which made the paper more concise.

An equivalent condition for the existence of a measurable cardinal.

Introduction. The main purpose of this paper is to prove the equivalence of the existence of a topological space in which there is a nonconvergent open ultrafilter with the countable intersection property and the existence of a measurable cardinal. This paper will answer indirectly the open question in [2 ]. 1.1 DEFINITION. We call a cardinal m measurable if a set X of cardinal m admits a { 0, 1 } valued measure ,u such that ,u(X) =1 and ,u({x }) = 0 for every xEX. For any set X, I XI will denote the cardinal of X. 1.2 DEFINITION. An open filter is a nonempty collection of open sets cU such that (1) 0 El, and (2) if U, VEza and G=int(G) D UnV, then G Ez A. An open ultrafilter is an open filter which is maximal in the collection of open filters.