scholarly journals A characterization of pseudocompactness

1981 ◽  
Vol 4 (2) ◽  
pp. 407-409
Author(s):  
Prabduh Ram Misra ◽  
Vinodkumar
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Hausdorff Space ◽  
Completely Regular ◽  
Pseudocompact Space

It is proved here that a completely regular Hausdorff space X is pseudocompact if and only if for any continuous function f from X to a pseudocompact space (or a compact space) Y, f*ϕ is z-ultrafilter whenever ϕ is a z-ultrafilter on X.

scholarly journals On -Cogenerated Commutative Unital -Algebras

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Ehsan Momtahan
Keyword(s):  
Compact Space ◽  
Commutative Algebra ◽  
Cardinal Number ◽  
Continuous Functions ◽  
Regular Space ◽  
Completely Regular ◽  
Simple Characterization ◽  
Maximal Ideals ◽  
Complex Valued

Gelfand-Naimark's theorem states that every commutative -algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space , is dense--separable if and only if is -cogenerated if and only if every family of maximal ideals of with zero intersection has a subfamily with cardinal number less than and zero intersection. This gives a simple characterization of -cogenerated commutative unital -algebras via their maximal ideals.

scholarly journals WEIGHTED COMPOSITION OPERATORS AND DYNAMICAL SYSTEMS

1998 ◽  
Vol 29 (2) ◽  
pp. 101-107
Author(s):  
R. K. SINGH ◽  
BHOPINDER SINGH
Keyword(s):  
Hausdorff Space ◽  
Weighted Space ◽  
Composition Operators ◽  
Continuous Functions ◽  
Weighted Composition Operators ◽  
Linear Dynamical System ◽  
Completely Regular ◽  
Weighted Composition ◽  
Locally Convex

Let $X$ be a completely regular Hausdorff space, $E$ a Hausdorff locally convex topo­logical vector space, and $V$ a system of weights on $X$. Denote by $CV_b(X, E)$ ($CV_o(X, E)$) the weighted space of all continuous functions $f : X \to E$ such that $vf (X)$ is bounded in $E$ (respectively, $vf$ vanishes at infinity on $X$) for all $v \in V$. In this paper, we give an improved characterization of weighted composition operators on $CV_b(X, E)$ and present a linear dynamical system induced by composition operators on the metrizable weighted space $CV_o(\mathbb{R}, E)$.

An Algebraic Characterization of Remainders of Compactifications

1983 ◽  
Vol 26 (3) ◽  
pp. 347-350
Author(s):  
James Hatzenbuhler ◽  
Don A. Mattson ◽  
Walter S. Sizer
Keyword(s):  
Hausdorff Space ◽  
Metric Spaces ◽  
Quotient Ring ◽  
Algebraic Characterization ◽  
Locally Compact ◽  
Completely Regular ◽  
Compact Metric Spaces ◽  
Primitive Idempotents

AbstractLet X be a locally compact, completely regular Hausdorff space. In this paper it is shown that all compact metric spaces are remainders of X if and only if the quotient ring C*(X)/C∞(X) contains a subring having no primitive idempotents.

scholarly journals On Hausdorff compactifications of non-locally compact spaces

1979 ◽  
Vol 2 (3) ◽  
pp. 481-486
Author(s):  
James Hatzenbuhler ◽  
Don A. Mattson
Keyword(s):  
Hausdorff Space ◽  
Locally Compact ◽  
Completely Regular ◽  
Compact Spaces ◽  
Hausdorff Compactifications ◽  
Set Of Points ◽  
Locally Compact Spaces

LetXbe a completely regular, Hausdorff space and letRbe the set of points inXwhich do not possess compact neighborhoods. AssumeRis compact. IfXhas a compactification with a countable remainder, then so does the quotientX/R, and a countable compactificatlon ofX/Rimplies one forX−R. A characterization of whenX/Rhas a compactification with a countable remainder is obtained. Examples show that the above implications cannot be reversed.

On the Katětov and Stone-Čech Extensions

1959 ◽  
Vol 2 (1) ◽  
pp. 1-4 ◽  
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Compact Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Approximation Theorem ◽  
General Topology ◽  
Natural Question ◽  
Completely Regular ◽  
Extension Spaces ◽  
Weierstrass Approximation Theorem

In general topology, one knows several standard extension spaces defined for one class of spaces or another and it is a natural question concerning any two such extensions which are defined for the same space whether they can ever be equal to each other. In the following, this problem will be considered for the Stone-Čech compactification βE of a completely regular non-compact Hausdorff space E[4] and Katětov's maximal Hausdorff extension κE of E[5]. It will be shown that βEκE always holds or, what amounts to the same, that κE can never be compact. As an application of this it will be proved that any completely regular Hausdorff space is dense in some non-compact space in which the Stone-Weierstrass approximation theorem holds.

scholarly journals 0-dimensional compactifications and Boolean rings

1968 ◽  
Vol 8 (4) ◽  
pp. 755-765 ◽  
Author(s):  
K. D. Magill ◽  
J. A. Glasenapp
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Hausdorff Space ◽  
Dimensional Space ◽  
Continuous Mapping ◽  
Dense Subspace ◽  
Clopen Subset ◽  
Partially Order ◽  
Completely Regular ◽  
Boolean Rings

A subset of a topological space which is both closed and open is referred to as a clopen subset. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. Here, a 0-dimensional space is a Hausdorff space which has a basis of clopen sets. By a compactification αX of a completely regular Hausdorff space X, we mean any compact space which contains X as a dense subspace. Two compactifications αX and γX are regarded as being equivalent if there exists a homeomorphism from αX onto γX which keeps X pointwise fixed. We will not distinguish between equivalent compactifications. With this convention, we can partially order any family of compactifications of X by defining αX ≧ γX if there exists a continuous mapping from γX onto αX which leaves X pointwise fixed. This paper is concerned with the study of the partially ordered family [X] of all 0-dimensional compactifications of a 0-dimensional space X.

Developability and Some New Regularity Axioms

1981 ◽  
Vol 33 (3) ◽  
pp. 641-663 ◽  
Author(s):  
N. C. Heldermann
Keyword(s):  
Continuous Function ◽  
Closed Subset ◽  
Natural Extension ◽  
Unit Interval ◽  
Regularity Conditions ◽  
Regular Space ◽  
Topological Spaces ◽  
Completely Regular ◽  
Natural Way

In a recent publication H. Brandenburg [5] introduced D-completely regular topological spaces as a natural extension of completely regular (not necessarily T1) spaces: Whereas every closed subset A of a completely regular space X and every x ∈ X\A can be separated by a continuous function into a pseudometrizable space (namely into the unit interval), D-completely regular spaces admit such a separation into developable spaces. In analogy to the work of O. Frink [16], J. M. Aarts and J. de Groot [19] and others ([38], [46]), Brandenburg derived a base characterization of D-completely regular spaces, which gives rise in a natural way to two new regularity conditions, D-regularity and weak regularity.

scholarly journals A note on zero-sets in the Stone-Čech compactification

1975 ◽  
Vol 12 (2) ◽  
pp. 227-230 ◽  
Author(s):  
D. Rudd
Keyword(s):  
Hausdorff Space ◽  
Completely Regular ◽  
Zero Sets

The ring C(X) is the ring of all continuous real-valued functions on a completely regular Hausdorff space X, and βX is the Stone-Čech compactification of X.The author proves a theorem which leads to a characterization of those zero-sets in X whose closures (in βX) are zero-sets in βX, and relates this characterization to the ideals in the ring C(X).

scholarly journals A characterization of a map whose inverse limit is an arc

2021 ◽  
pp. 1-17
Author(s):  
SINA GREENWOOD ◽  
SONJA ŠTIMAC
Keyword(s):  
Continuous Function ◽  
Inverse Limit ◽  
Splitting Sequence

Abstract For a continuous function $f:[0,1] \to [0,1]$ we define a splitting sequence admitted by f and show that the inverse limit of f is an arc if and only if f does not admit a splitting sequence.

The coincidence approach to stochastic point processes

1975 ◽  
Vol 7 (1) ◽  
pp. 83-122 ◽  
Author(s):  
Odile Macchi
Keyword(s):  
Poisson Process ◽  
Probability Space ◽  
Point Processes ◽  
Counting Process ◽  
Regular Point ◽  
General Point ◽  
Completely Regular ◽  
Photon Process ◽  
Stochastic Point Processes

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Sign in / Sign up

Export Citation Format

Share Document