A nice subclass of functionally countable spaces

1. 1. , -

Nouns: plurals, countable versus uncountable

Closed operator ideals on the Banach space of continuous functions on the first uncountable ordinal

The aim of this article is to modify the notion of the γ-property so that the modified property is preserved under taking a direct topological sum. To this end, we use a procedure of “saturation” of ω-covers of topological spaces. Namely, we add to each ω-cover the family of all unions of its subfamilies consisting of at most k elements (“k-saturation”), or the family of all unions of its finite subfamilies (“saturation”). Thus, we obtain for Tychonoff spaces a sequence ( k )k<ω γ′ of covering properties such that k k +1 ′ ′ γ ⇒ γ for any integer k, and 1 ′γ is the well-known γ-property. Also, the property ωγ′ such that k ω γ′ ⇒ γ′ for any integer k is obtained. It is proved that each kγ′ -property is preserved under usual topological operations, namely, taking closed subspaces, continuous images, and finite powers. It is known that the classical γ-property can be failed under passing to topological sum. Our main result means that it is impossible with respect to whole sequence ( k )k<ω γ′ . More precisely, if a space X satisfies kγ′ , Y satisfies mγ′ , then the sum X ⊕Y satisfies k +m γ′ . Nevertheless, X ⊕ X satisfies kγ′ as X itself. As another main result, we establish that the ωγ′ -property and the Lindelöf property are equivalent. It follows that any countable union of spaces Xn with the k (n) γ′ -property is a Lindelöf space.

The transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

We introduce the notion of a cellular-compact space and prove that cellular compactness is a nice property that implies cellular Lindelofness. The class of cellular-compact spaces is preserved by continuous images and finite unions, as well as by regular closed subsets and extensions. It is established that cellular-compact spaces must be pseudocompact but not necessarily countably compact. We prove that first countable cellular-compact regular spaces are countably compact and their cardinality does not exceed $${2^{\omega}}$$ . We also show that a collectionwise normal Frechet–Urysohn cellular-compact space need not be compact and there exist Frechet–Urysohn cellular-compact spaces that do not have a dense countably compact subspace.

Appendix C. Countable and Uncountable Sets

The theory of counting, or enumeration, is the theoretical counterpart of everyday practical counting. The chapter starts by defining an experiment to be any activity with well-defined, observable outcomes or results, and an event as a set of outcomes. Tree diagrams are introduced to represent sequences of events.The multiplication principle and the rule of sum are introduced. Venn diagrams are used to describe events and in counting the number of outcomes in unions and intersections of events. The principle of inclusion and exclusion is introduced.One-to-one correspondences are discussed, as are finite and infinite and countable and uncountable sets. Selections and arrangements are studied, leading to the binomial theorem.KeywordsVenn DiagramBritish Airways FlightTrack TeamPositive Integer IndexScrabble TilesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Continuous imaging of cardiac functions is highly desirable for the assessment of long-term cardiovascular health, detection of acute cardiac dysfunction and clinical management of critically ill or surgical patients1–4. However, conventional non-invasive approaches to image the cardiac function cannot provide continuous measurements owing to device bulkiness5–11, and existing wearable cardiac devices can only capture signals on the skin12–16. Here we report a wearable ultrasonic device for continuous, real-time and direct cardiac function assessment. We introduce innovations in device design and material fabrication that improve the mechanical coupling between the device and human skin, allowing the left ventricle to be examined from different views during motion. We also develop a deep learning model that automatically extracts the left ventricular volume from the continuous image recording, yielding waveforms of key cardiac performance indices such as stroke volume, cardiac output and ejection fraction. This technology enables dynamic wearable monitoring of cardiac performance with substantially improved accuracy in various environments.

4 - Measurable sets: The Carathéodory characterization

Locally compact, locally countable spaces and random reals

Let β N \beta N be the Stone-Čech compactification of the integers N. It is shown that p is a P-point of β N − N \beta N - N , then β N − N − { p } \beta N - N - \{ p\} is not normal. Let D be an uncountable discrete set and E 0 {E_0} be the set of points in β D − D \beta D - D in the closures of countable subsets of D It is shown that there is a two-valued continuous function on E 0 {E_0} which cannot be extended continuously to β D \beta D .

A new class of functions, called slightly generalized β-continuous functions is introduced.Basic properties of slightly generalized β-continuous functions are studied.The class of slightly generalized βcontinuous functions properly includes the class of slightly β-continuous functions and generalized β-continuous functions.Also, by using slightly generalized β-continuous functions, some properties of domain/range of functions are characterized.

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.

Dyadicity index and metrizability of compact continuous images of function spaces