Realcompact Research Articles

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf( c)>ω 1, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2 x of closed subsets of X is then not realcompact, and the extension μ f( vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.

Erratum to "A Maximal Realcompactification with 0-Dimensional Outgrowth"

Erratum to "A Maximal Realcompactification with 0-Dimensional Outgrowth"

Lattice repleteness and some of its applications to topology

We deal with the general concept of lattice repleteness. Specifically, we systematize the study of several important special cases of repleteness, namely, realcompactness, α-completeness, N-compactness, and Borel-completeness; we apply our general results on repleteness to specific lattices in topological spaces, in particular, to analytic spaces; we utilize the concept of G δ-closure to obtain necessary or sufficient conditions for repleteness (this portion of our work generalizes important theorems of Mrówka on Stone-Čechcompactification, of Frolik on realcompact spaces, and of Wenjen on realcompact spaces); finally, we extend the measure representation material of Varadarajan and then we utilize the results to obtain further applications to repleteness.

Erratum to: “A maximal realcompactification with 0-dimensional outgrowth” [Proc. Amer. Math Soc. 51 (1975), 441–447; MR 51#11435

Erratum to: “A maximal realcompactification with 0-dimensional outgrowth” [Proc. Amer. Math Soc. 51 (1975), 441–447; MR 51#11435

On a problem concerning generalizations of realcompact spaces

PF-compact spaces are defined. Every almost realcompact space is PF-compact and every separable PF-compact space is almost realcompact. Nonetheless in the constructible universe V = L, a space of cardinality ℵ 1 is PF-compact if, and only if, the closure of each of its countable subsets is almost realcompact, and ω 1 is hereditarily PF-compact.

An Embedding Characterization of Almost Compact Spaces

We characterize almost compact and almost realcompact spaces in terms of their situation in the product ${(J,u)^C}$. In the characterization of almost compactness $J$ is the two point set or the unit interval; in the characterization of almost realcompactness $J$ is the set of nonnegative integers or the nonnegative reals. $u$ is the upper topology on the real line restricted to $J$.

A space which contains no realcompact dense subspace

A Tychonoff space which contains no realcompact space as a dense subspace is constructed. Let D R \mathcal {D}\mathcal {R} be the class of all spaces which contain some realcompact spaces as dense subspaces. Then, as a consequence of the above result, it follows that D R \mathcal {D}\mathcal {R} is not closed-hereditary.

An embedding characterization of almost compact spaces

We characterize almost compact and almost realcompact spaces in terms of their situation in the product ( J , u ) C {(J,u)^C} . In the characterization of almost compactness J J is the two point set or the unit interval; in the characterization of almost realcompactness J J is the set of nonnegative integers or the nonnegative reals. u u is the upper topology on the real line restricted to J J .

A note on 𝛼-compact spaces

For an infinite cardinal α \alpha , m ( α ) m(\alpha ) denotes the least measurable cardinal, if one exists, not less than α \alpha . We give easy proofs of generalizations of some results on realcompact spaces. Among these we prove the following generalization of a theorem of A. Kato. Let { X i : i ∈ I } \{ {X_i}:i \in I\} be a collection of spaces each having at least two elements. Then the k k -box product ( ∏ X i ) k {(\prod {X_i})_k} is α \alpha -compact if and only if either X i {X_i} is α \alpha -compact for each i ∈ I i \in I and k ⩽ m ( α ) k \leqslant m(\alpha ) or | I | > m ( α ) \left | I \right | > m(\alpha ) .

Versetzungsaktivierte schwindungsvorgänge beim einkomponenten-sintern

Mechanisms leading to large amounts of shrinkage and high shrinkage rates during sintering of powder compacts are discussed. The underlying ideas are based upon the experimental observation on the formation of high dislocation densities in regions close to the contact grain boundary. The dislocation structure in these regions is relatively stable during technical sintering times. It is shown that, in addition to effects on diffusional high-temperature creep, the rearrangement of powder particles or groups of particles by tilting and gliding is activated and accelerated due to the enrichment of dislocations in the contact zones. Since these rearrangement processes are highly effective with respect to shrinkage and require relatively minor fluxes, they are important for understanding the shrinkage of real powder compacts.

Realcompactness and the Notion of Observable

V. S. Varadarajan proved in [7] that any classical observable on the real line is a measurable mapping. We show in the present note that such a statement holds true for all realcompact spaces of nonmeasurable cardinality. This we think may find applications in the realm of quantum mechanics. Moreover, since realcompactness is always necessary our result may be viewed as a characterization of realcompact spaces (with the usual reservation for measurable cardinals).

One-dimensional linear coherent processing using a single optical element

A coherent processor is presented which is capable of performing a large class of 1-D linear space-variant operations. The only components of the processor are a 1-D input, a mask whose transmittance is specified by the desired linear operation, and an output plane. Compared with other 1-D processors, this processor has advantages of real space compactness and total elimination of vignetting. Experimental results are presented for the specific operations of convolution and spectrum scaling.

The category of all zero-dimensional realcompact spaces is not simple

For every zero-dimensional space E of non-measurable cardinality we construct a zero-dimensional, hereditarily realcompact, locally compact and locally countable space which cannot be embedded as a closed subspace into any topological power of the space E. Under the assumption that all cardinals are non-measurable it gives the result stated in the title. This is an answer for a question raised by H. Herrlich

Topologically complete spaces and perfect maps

1.IntrodllCtion.A11spacesconsideredin this paper are assumed to be completelyregular Hausdorff.A space Xis ca11ed tqi>Ologicalcthis fact was essentially provedby Mr6wka[18]and was notedin[3].To date,theimages of realcompact spaces under perfect maps wereinvestigated by severaltopologists(e.g.,Frolik[6],[7],Kenderov[14],Isiwata [11],[13],Blair[1],Dykes[3],[4]),however,With the exception of[3],1ittle seems to beknown about topologically complete spaces. In this paper,We Shallobtain characterizations of theimages of topologlCally COmplete spaces under perfect mapsandnecessaryandsu疏cientconditionsforthem to be topologically complete. In section2,for convenience,Welist certain basic de丘nitions and facts that Willbe usedin the sequal. In section3,Weintroduce the notionofalmostuniforlnStruCtureS.ThisllOtion isusefulfordealingwiththeperfectimages of topologlCally complete spaces.There are also some“tool”theorems concernlng almost uniform struCtureS. In section4,almost topological1y complete spaces are de丘nedin tel・mS Of an almost uniform structure,and we prove that almost topologlCa11y complete spaces Characterizeperfectimagesoftopologically completespaces.Similarly we can prove the corresponding theorem concernlng Frolik’s almost realcompactspaces,and conSider the relationship between almost topologlCally complete spaces and almost re・ alcompact spaces・Ful-thermore,SOme prOperties of almost topologlCallycomplete SPaCeS are Studied,in particular,itis proved that almost topologlCalcompleteness isinvariant under perfect maps. In the負nalsection5,We COnSider a problem underwhat conditionsan almost

𝑁-compactness and weak homogeneity

A characterization of those locally compact, 0-dimensional, realcompact spaces X that are N-compact is given in terms of a density condition on β X − X \beta X - X .

Extensions of completely regular ordered spaces

In this paper we introduce the concept of o-completely regular filters on a completely regular ordered space in order to characterize compact ordered spaces and show that for any completely regular ordered space X, Nachbin compactification of X is precisely the strict extension of X with all maximal o-completely regular filters as the filter trace. With this concept, we also define k-compact ordered spaces for every infinite cardinal /c, which give rise to a chain of extensive subcategories of the category of completely regular ordered spaces, analogous to the chain given by the category of k-compact spaces, for the different cardinals k. 0. Introduction. Nachbin [15] has introduced the concept of completely regular ordered space as a generalization of completely regular topological space and, as a generalization of the Stone-Cech compactification, has shown that the category of compact ordered spaces and continuous, isotones is an extensive subcategory (2.2) of the category of completely regular ordered spaces and continuous isotones. Herrlich [8] has introduced k -compact spaces for each infinite cardinal and has shown that they form a chain of extensive subcategories of the category of completely regular spaces and continuous maps. Introducing o-completely regular filters on a completely regular ordered space, it is shown that for a completely regular ordered space X, Nachbin compactification of X is the strict extension of X with all maximal o-completely regular filters as the filter trace. Using this fact, we define k -compact ordered spaces for every infinite cardinal fc, and show that categories of k -compact ordered spaces are extensive in the category of completely regular ordered spaces and continuous isotones. It is known [11] that a completely regular space is k -compact iff it is k-closed (3.2) in its Stone-Cech compactification and Krcompact spaces are precisely realcompact spaces, or closed subspaces of powers of the real line R. In our case, k -compact ordered spaces are precisely those spaces which are k -closed in their Nachbin compactifications, but it is shown that Nrcompact ordered space need not be an R -compact ordered space, i.e. a closed subspace of a power of R. Finally it is noted that by replacing continuous isotones between topological partially ordered spaces with continuous homomorphisms

On hard sets

Hard sets, a class between compact sets and realcompact closed sets, are introduced and some of their properties are studied. As an application, a necessary and sufficient condition for the quotient of a realcompact space to be realcompact is found in terms of hard sets.

Pseudocompactness properties

A topological extension property is a class of Tychonoff spaces P \mathcal {P} which is closed hereditary, closed under formation of topological products and contains all compact spaces. If X X is Tychonoff and P \mathcal {P} is an extension property, there is a space P X \mathcal {P}X such that X ⊆ P X ⊆ β X , P X ∈ P X \subseteq \mathcal {P}X \subseteq \beta X,\;\mathcal {P}X \in \mathcal {P} and if f : X → Y f:X \to Y where Y ∈ P Y \in \mathcal {P} then f f admits a continuous extension to P X \mathcal {P}X . A space X X is called P \mathcal {P} -pseudocompact if P X = β X \mathcal {P}X = \beta X . In this note it is shown that if P \mathcal {P} is an extension property which contains the real line (e.g., the class of realcompact spaces), X X is P \mathcal {P} -pseudocompact and Y Y is compact, then X × Y X \times Y is P \mathcal {P} -pseudocompact. An example is given of an extension property P \mathcal {P} , a P \mathcal {P} -pseudocompact space X X and a compact space Y Y such that X × Y X \times Y is not P \mathcal {P} -pseudocompact.

Prediction of Soil Compaction Due to Off-Road Vehicle Traffic

ABSTRACT SIMULTANEOUS field and iab-*J) oratory compaction data were examined in order to determine the prediction possibilities. Several existing theories were also considered for comparison. An equation repre-senting dry density in terms of applied pressure and moisture content was used to describe the field and labor-atory data. The pressure applica-tion on the agricultural soil varied from near 0 to 1.62 kg/cm2, and field data obtained at various moisture contents provided necessary informa-tion for arriving at the laboratory index test methods. The pattern of real compaction under tire loads was found to compare well with the standard Proctor compaction test. Results of such attempts are very useful for off-road machninery de-signers and farmers.

A Maximal Realcompactification with 0-Dimensional Outgrowth

A Maximal Realcompactification with 0-Dimensional Outgrowth