Weak Partition Research Articles

Improved adeno-associated virus empty and full capsid separation using weak partitioning multi-column AEX chromatography.

Recombinant adeno-associated virus (rAAV) empty and full capsid separation has been a topic of interest in the rAAV gene therapy community for many years and the anion exchange chromatography (AEX) step has undergone various process optimizations to improve rAAV empty capsid separation, including AEX stationary phase, mobile phase, and process parameters. Here, we present a new AEX method that employs both weak partitioning chromatography (WPC) and multi-column chromatography (MCC) to achieve improved full rAAV percentage in the AEX pool. The WPC technology allows empty rAAV to be displaced by full rAAV during loading, while the MCC technology enables parallel column processing which further increases AEX step productivity. Our results show that, compared to baseline AEX batch chromatography, the AEX-WPC-MCC method demonstrated improvements in both AEX pool full rAAV percentage (∼ 20% increase) and rAAV genome recovery (∼ 20% increase). As a result, the productivity (full capsid generated per liter of AEX column per hour of processing time) of the AEX step increased by ∼34-fold from the baseline AEX batch run to the AEX-WPC-MCC run. It is foreseeable that this AEX-WPC-MCC method could find applications in large-scale rAAV manufacturing processes to improve AEX yield and reduce the cost of goods of rAAV manufacturing.

Characterization of synthetic guide ribonucleic acids through hydrophilic interaction chromatography coupled with mass spectrometry

In this study, we aimed to develop a hydrophilic interaction liquid chromatography (HILIC) method for the analysis of single guide ribonucleic acid (sgRNA), a critical reagent used in CRISPR genome editing. Our results showed that effective profiling of sgRNA can be achieved by suppressing the surface charge of the stationary phase in HILIC. We identified hydrogen bonding as the primary retention mechanism with potential weak partitioning in HILIC separation of large oligonucleotides like 100-mer sgRNA. Moreover, we demonstrated that direct coupling of HILIC with mass spectrometry (MS) allows the intact mass analysis of sgRNA and its impurities with minimal adduct present. Finally, we characterized the post peak shown in the low temperature HILIC and identified it as sgRNA aggregates. Our findings provide valuable insight into the characterization of sgRNA and highlight the potential of HILIC-MS as a powerful analytical tool for relatively large oligonucleotide analysis.

Effectively removing the homodimer in bispecific antibodies by weak partitioning mode of anion exchange chromatography

Small amounts of by-products are nevertheless created during the recombinant production of IgG-like bispecific antibodies due to imbalanced chain expression and improper chain pairing, despite the employment of molecular strategy techniques to promote accurate pairing. Among them, homodimers represent the species that are more difficult to remove due to their physical and chemical properties being similar to the target antibody. Homodimer by-products are always produced even though various technologies can significantly increase the expression of heterodimers, so a robust purification process to recover high-purity heterodimers is required. Most of the chromatography methods commonly adopt the bind-and-elute mode or two-step to separate homodimers, which has numerous drawbacks such as prolonged process times and limited dynamic binding capacity. Flow-through mode of anion exchange is a frequently-used polishing step for antibodies, but it is typically regarded as being more effective for host-cell protein or host-cell DNA removal rather than other product-related impurities such as homodimers and aggregates. This paper demonstrated that single-step anion exchange chromatography allows high capacity and effective clearance of the homodimer byproduct to be simultaneously achieved, suggesting that weak partitioning was a better polishing strategy for achieving a high level of heterodimer purity. And robust operation range of anion exchange chromatography steps for homodimer removal was also developed by leveraging the design of experiments.

Seasonal Oxygenation of Contaminated Floodplain Soil Releases Zn to Porewater.

Soil zinc contamination is a major threat to water quality and sensitive ecosystems. While Zn itself is not redox-active in soils, transitions in soil redox conditions may promote mobilization of Zn from common Zn hosts, including Mn(IV)/Fe(III)-(hydr)oxides and sulfide precipitates, leading to elevated concentrations of dissolved Zn in surface and groundwater and thus a potential increase in Zn transport and uptake. Here, we examined the impacts of hydrologic fluctuations and coupled redox transitions on Zn partitioning in contaminated riparian soil in a mountain watershed. We found that oxygenation of the soil profile during low water conditions caused a spike in porewater Zn concentrations, driven by oxidative dissolution of amorphous ZnS and weak partitioning of Zn to Fe(III)-(hydr)oxides, hydroxy-interlayer vermiculite, and vermiculite. In contrast to Pb, released Zn did not immediately adsorb to Fe(III)-(hydr)oxides or particulate organic matter due to less-favorable sorption of Zn than that of Pb and, further, decreased Zn sorption at slightly acidic pH. As aridification intensifies and groundwater levels decline throughout the western United States, contaminated floodplain soils in mountain watersheds may be frequently oxygenated, leading to increased mobilization of dissolved Zn, which will amplify the threat Zn poses to water quality and ecosystem health.

Rapid Multicomponent Alloy Solidification with Allowance for the Local Nonequilibrium and Cross-Diffusion Effects.

Motivated by the fast development of various additive manufacturing technologies, we consider a mathematical model of re-solidification of multicomponent metal alloys, which takes place after ultrashort (femtosecond) pulse laser melting of a metal surface. The re-solidification occurs under highly nonequilibrium conditions when solutes diffusion in the bulk liquid cannot be described by the classical diffusion equation of parabolic type (Fick law) but is governed by diffusion equation of hyperbolic type. In addition, the model takes into account diffusive interaction between different solutes (nonzero off-diagonal terms of the diffusion matrix). Numerical simulations demonstrate that there are three main re-solidification regimes, namely, purely diffusion-controlled with solute partition at the interface, partly diffusion-controlled with weak partition, and purely diffusionless and partitionless. The type of the regime governs the final composition of the re-solidified material, and, hence, may serve as one of the main tools to design materials with desirable properties. This implies that the model is expected to be useful in evaluating the most effective re-solidification regime to guide the optimization of additive manufacturing processing parameters and alloys design.

More definable combinatorics around the first and second uncountable cardinals

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text].

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of \(\sigma \)-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter \({\mathcal {U}}\) for n-tuples, denoted \(t({\mathcal {U}},n)\), is the smallest number t such that given any \(l\ge 2\) and coloring \(c:[\omega ]^n\rightarrow l\), there is a member \(X\in {\mathcal {U}}\) such that the restriction of c to \([X]^n\) has no more than t colors. Many well-known \(\sigma \)-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at best. In this paper, we utilize methods of topological Ramsey spaces to calculate Ramsey degrees of several classes of ultrafilters generated by \(\sigma \)-closed forcings. These include a hierarchy of forcings due to Laflamme which generate weakly Ramsey and weaker rapid p-points, forcings of Baumgartner and Taylor and of Blass and generalizations, and the collection of non-p-points generated by the forcings \({\mathcal {P}}(\omega ^k)/\mathrm {Fin}^{\otimes k}\). We provide a general approach to calculating the Ramsey degrees of these ultrafilters, obtaining new results as well as streamlined proofs of previously known results. In the second half of the paper, we calculate pseudointersection and tower numbers for these \(\sigma \)-closed forcings and their relationships with the classical pseudointersection number \({\mathfrak {p}}\).

Removing a light-chain missing byproduct by MMC ImpRes mixed-mode chromatography under weak partitioning mode in purifying a WuXiBody-based bispecific antibody

WuXiBody is a bispecific antibody (bsAb) platform developed by WuXi Biologics. Its key feature is the replacement of one parental antibody's CH1/CL region with the T cell receptor (TCR) constant domain, a design that promotes cognate heavy chain (HC)-light chain (LC) pairing. BsAbs based on WuXiBody can adopt either asymmetric or symmetric format. For purifying a WuXiBody-based symmetric bsAb, we identified a LC-missing species as a major byproduct. While for bsAbs based on other platforms removal of such byproduct can pose considerable challenge to the downstream team, in this case WuXiBody's unique design makes separation relatively straightforward. We previously showed that Capto MMC ImpRes mixed-mode chromatography under bind-elute mode can effectively remove this LC-missing species. However, the dynamic binding capacity (DBC) of Capto MMC ImpRes is relatively low under the selected condition, making the process less desirable for large-scale manufacturing. In this study, we demonstrated that when Capto MMC ImpRes chromatography is conducted under weak partitioning mode, high throughput, good yield, and effective byproduct removal are simultaneously achieved.

Removal of host cell proteins from cell culture fluids by weak partitioning chromatography using peptide-based adsorbents

This work presents the removal of host cell proteins (HCPs) from a Chinese Hamster Ovary clarified cell culture fluid (CHO CCCF) containing a therapeutic monoclonal antibody (mAb) by weak partitioning chromatography (WPC). The chromatographic adsorbents were produced by functionalizing Toyopearl resin with HCP-binding tetrameric multipolar (4MP) or hexameric hydrophobic/cationic (6HP) peptides. The CCCF was loaded on columns packed with either 4MP-Toyopearl or 6HP-Toyopearl resin only, or a 4MP and 6HP resin mixture at different values of residence time (RT: 0.5, 1, 2, and 5 min). The temporal profiles of concentration of HCPs and mAb in the effluents confirmed the binding mechanism by WPC, where both HCPs and mAb are initially bound by the peptide ligands, but, as more CCCF is fed to the column, the incoming HCPs displace the bound mAbs. In particular, 4MP was shown to capture more selectively high molecular weight HCPs, while 6HP was more effective in binding low molecular weight HCPs. Under optimal loading conditions (~60–80 g of proteins per L of adsorbent; RT of 5 min), the 6HP+4MP-Toyopearl adsorbent provided mAb yield and purity of >80% and up to 90%, respectively. Conversely, the control resin Toyopearl SuperQ-650 M resulted in 70% yield and 75% purity under the same conditions. Proteomic analysis of the effluents demonstrated that 6HP+4MP-Toyopearl adsorbent removes HCPs known for their immunogenicity or IgG co-elution or degradation, demonstrating the potential of these peptide-based resins as HCP scrubbers in mAb purification processes.

On The Strong and the Semi-Strong Path Partition Conjecture

We define two families of graphs and prove the strong and the weak path partition conjectures for subfamilies of these graphs.

The effect of pH on antibody retention in multimodal cation exchange chromatographic systems

The present paper builds upon previous work on mAb domain contributions to multimodal (MM) chromatography by examining how pH can impact mAb surface properties and retention in these systems. Linear salt gradient experiments were carried out between pH 5–7 for several mAbs with different pI and surface hydrophobicities in four different MM CEX resins at two ligand densities. mAb retention showed an inverse, non-linear correlation with pH. Changing pH affected the elution order, creating unique windows of selectivity in each of the MM CEX resins. One mAb showed a pH-dependent spectrum of domain contributions, demonstrating that pH can be used to tune the relative importance of the (Fab)2 and Fc domains for some mAbs in MM systems. Positive, negative, and hydrophobic patches were calculated between pH 5–7 for the mAbs. Visualizing these patches on the protein surface demonstrated that each mAb showed a unique distribution of surface charge and hydrophobicity that changed with pH. The sum of patch areas was tracked across this pH range to quantitatively understand how pH impacted these important surface properties. The quantitative analysis then was narrowed to consider only patches in the CDR loops, which were hypothesized to be an important interaction site for some mAbs in these systems. Interestingly, differences in the titration of CDR loop patches for each mAb were shown to be a result of Histidine titrations and patches in this region were qualitatively correlated with experimental trends including the observed elution order reversals. These results indicate that pH potentially can be employed as a lever for the strategic design of multimodal steps to create flow through, bind and elute, or weak partitioning operations with important implications for the design of integrated and/or continuous downstream purification processes. Furthermore, the ability to tune domain contributions in MM separations using pH creates intriguing possibilities for current downstream challenges such as the removal of product-related impurities, as well as the purification of bispecific mAbs.

Weak niche partitioning by migrating shorebirds in a single-food-type environment

This study examined foraging niche partitioning among coexisting species of shorebirds on a sandy beach dominated by a single food type, Horseshoe Crab ( Limulus polyphemus) eggs, where use of foraging microhabitats of limited variety is temporarily restricted due to tidal cycle. The major goal of the study was to examine whether co-occurring species differed in selection of foraging microhabitats and use of feeding techniques. Ruddy Turnstone ( Arenaria interpres), Red Knot ( Calidris canutus), and Sanderling ( Calidris alba) foraged significantly more on sand and gravel compared to pools and swash zone, Semipalmated Sandpiper ( Calidris pusilla) foraged significantly more on sand and gravel compared to swash zone, but not significantly more compared to pools, while differences in use of microhabitats by Dunlin ( Calidris alpina), and Short-billed Dowitcher ( Limnodromus griseus) were non-significant. Turnstones used digging and turning stones, and pecking significantly more than probing, Semipalmated Sandpipers used pecking significantly more than probing, while all other species used probing significantly more than pecking. Knots and Sanderlings had the highest level of overlap both in use of microhabitats and foraging techniques, while turnstones and dowitchers had the lowest. Ruddy Turnstones were the only birds that used digging and turning stones to obtain food and these two foraging methods seemed to be the most important variables discriminating turnstones from other co-occurring shorebirds. However, despite the slight differences in use of foraging methods and microhabitats by different species, this study documented weak partitioning among shorebirds, particularly between sandpipers and dowitchers. These results indicate that the short-term absence of clear niche partitioning on a local scale may occur in shorebird communities.

Fluid-melt trace-element partitioning behaviour between evolved melts and aqueous fluids: Experimental constraints on the magmatic-hydrothermal transport of metals

The fluid-melt partition coefficients of ten trace metals (Li, Mn, Cu, Zn, Rb, Sr, Mo, W, and Pb, along with K) between hydrous evolved melts and coexisting Cl-bearing fluid(s) have been experimentally determined at 850–1050 °C, 0.6–2.2 kbar, and ƒO2 ≈ NNO –0.5 to NNO +2 log units. The measured compositions of the quenched fluid phases vary from low salinity (~2 wt% Cl−) to >19 wt% Cl−, with maximum starting fluid concentrations of >21 wt% Cl. The results of 15 partitioning experiments confirm that Dmetalfluid/melt values depend strongly on the melt and fluid composition, with the greatest metal affinities for the fluid phase shown in the experiments with high-Cl fluids, and the most fractionated melts. Five experiments conducted in Au96Cu4 capsule alloy show DCufluid/melt values ranging from ~40 to >80, and a strong correlation with melt FeO wt% contents at equivalent fluid Cl− contents. In contrast, Li shows only weak partitioning in favour of coexisting fluids across all melt compositions, with DLifluid/melt ranging from 0.16 in low salinity fluids to 1.17 in the highest Cl fluid (19.15 wt% Cl−).A procedure was employed to redissolve and analyse the aluminosilicate precipitates that form during quenching of the experiments. This procedure allows for metal contents of the final quench fluids to be constrained by both mass balance and direct measurement by solution ICP-MS analysis. A comparison of the mass balance calculated vs. directly measured fluid-melt partition coefficients generally gives good agreement, within an order of magnitude for most trace elements. However, discrepancies between these methods arise due to errors inherent in the mass balance calculations for metals with low abundances in the fluids (e.g. Sr), and especially those showing affinities to alloy with the experiment capsule material (e.g. Mo). These discrepancies in constraining experimental fluid trace element budgets also suggest that the precious metal capsule alloys cannot always be considered completely inert containers in experimental investigations, and semi-quantitative LA-ICP-MS analysis of the raw capsule alloys confirms the detectable presence of several of the trace elements investigated here.Integrating these new partitioning data with previous experimental and natural studies confirms that aqueous Cl-bearing fluids exsolved by hydrous calc-alkaline melts stored at mid- to upper-crustal conditions are capable of efficiently sequestering and transporting some economically important trace metals (e.g. Mn, Cu, Zn). However, the new fluid/melt Li partitioning data, in conjunction with evidence of very late (post-eruptive) Li modification, suggests that Li is moderately-strongly fluid-immobile at depth and suffers from significant post-eruptive re-equilibration. This has implications for deciphering the signatures of ‘volatile fluxing’ in arc magmas that are preserved in mineral and melt inclusion trace element concentrations, and the development of mineralised stockwork regions in porphyry-style metal deposits.

An axiomatic characterization of the Borda mean rule

A social dichotomy function maps a collection of weak orders to a set of dichotomous weak orders. Every dichotomous weak order partitions the set of alternatives into approved alternatives and disapproved alternatives. The Borda mean rule returns all dichotomous weak orders that approve all alternatives with above-average Borda score and disapprove alternatives with below-average Borda score. We show that the Borda mean rule is the unique social dichotomy function satisfying neutrality, reinforcement, faithfulness, and the quasi-Condorcet property. Our result holds for all domains of weak orders that are sufficiently rich, including the domain of all linear orders and the domain of all weak orders.

Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraisse classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokic is extended to equivalence relations for finite products of structures from Fraisse classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlak–Rodl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraisse classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not $$k+1$$ -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of n many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $$\mathcal {P}(n)$$ . If the number of Fraisse classes on each block grows without bound, then the Tukey types of the p-points below the space’s associated ultrafilter have the structure exactly $$[\omega ]^{<\omega }$$ . In contrast, the set of isomorphism types of any product of finitely many Fraisse classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.

Partitioning the vertex set of $G$ to make $G\,\Box\, H$ an efficient open domination graph

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.

Development of a modular virus clearance package for anion exchange chromatography operated in weak partitioning mode.

Anion exchange chromatography (AEX) operated under weak partitioning mode has been proven to be a powerful polishing step as well as a robust viral clearance step in Pfizer's monoclonal antibody (mAb) platform purification process. A multivariate design of experiment (DoE) study was conducted to understand the impact of operating parameters and feedstream impurity levels on viral clearance by weak partitioning mode AEX. Bacteriophage was used initially as a surrogate for neutral and acidic isoelectric point mammalian viruses (e.g., retrovirus and parvovirus). Five different mAbs were used in the evaluation of process parameters such as load challenge (both product and impurities), load pH, load conductivity, and contact time (bed height and flow-rate). The operating ranges obtained from phage clearance studies and Pfizer's historical data were used to define an appropriate operating range for a subsequent clearance study with model retrovirus and parvovirus. Both phage and virus clearance evaluations included feedstreams containing different levels of impurities such as high molecular mass species (HMMS), host cell proteins (HCPs), and host cell DNA. For all the conditions tested, over 5 log10 of clearance for both retrovirus and parvovirus was achieved. The results demonstrated that weak partitioning mode AEX chromatography is a robust step for viral clearance and has the potential to be included as part of the modular viral clearance approach.

A comparison of the diet and fine‐scale distribution of sympatric Tibetan and red foxes in Qinghai, PR China

We compared the diet and the spatial distribution of the Tibetan fox Vulpes ferrilata and the red fox Vulpes vulpes in the Tibetan plateau, to elucidate mechanisms of coexistence for these two sympatric canids and to clarify their roles as definitive hosts for zoonotic Echinococcus parasites. Diet and fine‐scale distribution patterns were assessed by fecal DNA analysis. A total of 45 fecal samples (15 belonging to Tibetan fox, 30 belonging to red fox were collected from 15 sites into three of which contained only Tibetan fox feces, six only red fox feces, and six contained feces of both species. The abundance of pika burrows, a key prey item for both species, did not differ among the sites. Food composition analysis, estimated using a point‐frame method, revealed slight but insignificant differences between the two species. Tibetan foxes consumed primarily mammals, whereas red foxes consumed primarily insects. The dietary range of the Tibetan fox was narrower than that of the red fox but there was little dietary overlap between the two species. These findings suggest that the weak partitioning of food resources between Tibetan and red foxes can facilitate their coexistence even within the same habitat where they share the same key prey items, i.e. small mammals such as pikas. These dietary differences between the two fox species also suggest that the Tibetan fox is a more important definitive host for Echinococcus on the Tibetan plateau than is the red fox.

A tandem laboratory scale protein purification process using Protein A affinity and anion exchange chromatography operated in a weak partitioning mode

A significant consequence of scaling up production of high titer monoclonal antibody (mAb) processes in existing facilities is the generation of in-process pools that exceed the capacity of storage vessels. A semi-continuous downstream process where columns and filters are linked and operated in tandem would eliminate the need for intermediate holding tanks. This study is a bench-scale demonstration of the feasibility of a tandem process for the purification of mAbs employing an affinity Protein A capture step, followed by a flow-through anion-exchange (AEX) step with the possibility of adding an in-line virus filtration step (VF). All three steps were linked sequentially and operated as one continuous process using an ÄKTA FPLC equipped with two pumps and a system of valves and bypasses that allowed the components to be engaged at different stages of the process. The AEX column was operated in a weak partitioning (WP) mode enabled by a precise in-line titration of Protein A effluent. In order to avoid complex control schemes and facilitate validation, quality and robustness were built into the system through selection of buffers based on thermodynamic and empirical models. The tandem system utilized the simplest possible combination of valves, pumps, controls, and automation, so that it could easily be implemented in a clinical or commercial production facility. Linking the purification steps in a tandem process is expected to generate savings in time and production costs and also reduce the size of quality systems due to reduced documentation requirements, microbial sampling, and elimination of hold time validation.

Weak partition properties on trees

We investigate the following weak Ramsey property of a cardinal ?: If ? is coloring of nodes of the tree ? \mathfrak{d}}$$ is regular, then $${{\kappa \rightsquigarrow (\kappa)^{ < \omega}_{\omega}}}$$ and that $${\mathfrak{b}}$$ $${(\mathfrak{b})^{ < \omega}_{\omega}}$$ and $${\mathfrak{d}}$$ $${(\mathfrak{d})^{ < \omega}_{\omega}}$$ . The arrow is applied to prove a generalization of a theorem of Hurewicz: A ?ech-analytic space is ?-locally compact iff it does not contain a closed homeomorphic copy of irrationals.