Cardinal functions of the hyperspace of convergent sequences

2018 ◽  
Vol 68 (2) ◽  
pp. 431-450 ◽  
Author(s):  
David Maya ◽  
Patricia Pellicer-Covarrubias ◽  
Roberto Pichardo-Mendoza
Keyword(s):  
Hausdorff Space ◽  
Current Paper ◽  
Vietoris Topology ◽  
Cardinal Function ◽  
Cardinal Functions ◽  
Lindelöf Number ◽  
Dispersion Character ◽  
Convergent Sequences

Abstract The symbol 𝓢c(X) denotes the hyperspace of all nontrivial convergent sequences in a Hausdorff space X. This hyperspace is endowed with the Vietoris topology. In the current paper, we compare the cellularity, the tightness, the extent, the dispersion character, the net weight, the i-weight, the π-weight, the π-character, the pseudocharacter and the Lindelöf number of 𝓢c(X) with the corresponding cardinal function of X. We also answer a question posed by the authors in a previous paper.

scholarly journals Hyperspace of finite unions of convergent sequences

2021 ◽  
Author(s):  
JingLing Lin ◽  
Fucai Lin ◽  
Chuan Liu
Keyword(s):  
Convergent Sequence ◽  
Vietoris Topology ◽  
Cardinal Function ◽  
Cardinal Invariants ◽  
Metric Properties ◽  
Generalized Metric ◽  
Network Weight ◽  
Hyper Space ◽  
Convergent Sequences

The symbol S(X) denotes the hyperspace of finite unions of convergent sequences in a Hausdor˛ space X. This hyper-space is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in S(X). Then we consider some cardinal invariants on S(X), and compare the character, the pseudocharacter, the sn-character, the so-character, the network weight and cs-network weight of S(X) with the corresponding cardinal function of X. Moreover, we consider rank k-diagonal on S(X), and give a space X with a rank 2-diagonal such that S(X) does not Gδ -diagonal. Further, we study the relations of some generalized metric properties of X and its hyperspace S(X). Finally, we pose some questions about the hyperspace S(X).

scholarly journals Cardinal Functions of Purely Atomic Measures

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
Szymon Gła̧b ◽  
Jacek Marchwicki
Keyword(s):  
Finite Measure ◽  
Topological Properties ◽  
Cardinal Function ◽  
Cardinal Functions

AbstractLet $$\mu $$ μ be a purely atomic finite measure. Without loss of generality we may assume that $$\mu $$ μ is defined on $${\mathbb {N}}$$ N , and the atoms with smaller indexes have larger masses, that is $$\mu (\{k\})\ge \mu (\{k+1\})$$ μ ( { k } ) ≥ μ ( { k + 1 } ) for $$k\in {\mathbb {N}}$$ k ∈ N . By $$f_\mu :[0,\infty )\rightarrow \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ f μ : [ 0 , ∞ ) → { 0 , 1 , 2 , ⋯ , ω , c } we denote its cardinal function $$f_{\mu }(t)=\vert \{A\subset {\mathbb {N}}:\mu (A)=t\}\vert $$ f μ ( t ) = | { A ⊂ N : μ ( A ) = t } | . We study the problem for which sets $$R\subset \{0,1,2,\dots ,\omega ,{\mathfrak {c}}\}$$ R ⊂ { 0 , 1 , 2 , ⋯ , ω , c } there is a measure $$\mu $$ μ such that $$R=\text {rng}(f_\mu )$$ R = rng ( f μ ) . We are also interested in the set-theoretic and topological properties of the set of $$\mu $$ μ -values which are obtained uniquely.

scholarly journals A Common Extension of Arhangel’skĭ’s Theorem and the Hajnal–Juhász Inequality

2020 ◽  
Vol 63 (1) ◽  
pp. 197-203 ◽  
Author(s):  
Angelo Bella ◽  
Santi Spadaro
Keyword(s):  
Hausdorff Space ◽  
Topological Spaces ◽  
Separation Axiom ◽  
Cardinal Invariants ◽  
Common Extension ◽  
Weak Lindelöf Number ◽  
Lindelöf Number

AbstractWe present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.

On the Linear Invariance of Lindelöf Numbers

1996 ◽  
Vol 39 (2) ◽  
pp. 129-137 ◽  
Author(s):  
Jan Baars ◽  
Helma Gladdines
Keyword(s):  
Topological Property ◽  
Dense Subset ◽  
Continuous Linear ◽  
Linear Bijection ◽  
Cardinal Functions ◽  
Lindelöf Number ◽  
Continuous Images

AbstractLet X and Y be Tychonov spaces and suppose there exists a continuous linear bijection from Cp(X)to CP(Y). In this paper we develop a method that enables us to compare the Lindelöf number of Y with the Lindelöf number of some dense subset Z of X. As a corollary we get that if for perfect spaces X and Y, CP(X) and Cp(Y)are linearly homeomorphic, then the Lindelöf numbers of Jf and Fare equal. Another result in this paper is the following. Let X and Y be any two linearly ordered perfect Tychonov spaces such that Cp(X)and Cp(Y)are linearly homeomorphic. Let be a topological property that is closed hereditary, closed under taking countable unions and closed under taking continuous images. Then X has isproperty if and only if Y has. As examples of such properties we consider certain cardinal functions.

Weak and norm sequential convergence in M(S)

1973 ◽  
Vol 15 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. Tong ◽  
D. Wilken
Keyword(s):  
Uniform Convergence ◽  
Compact Hausdorff Space ◽  
Dual Space ◽  
Hausdorff Space ◽  
Compact Operators ◽  
Borel Measures ◽  
Sequential Convergence ◽  
Continuous Complex ◽  
Complex Valued ◽  
Convergent Sequences

Let S be a compact Hausdorff space; let C(S) be the algebra of all continuous complex valued functions on S; and let M(S) be the dual space of (S) (the space of all regular Borel measures on S). In [2] Grothendieck gave a description of weak sequential convergence in M(S) in terms of uniform convergence on sequences of disjoint open sets in S. In this note we give a condition on the carriers of measures to guarantee that weak zero convergent sequences are norm zero convergent. While this condition is interesting in its own right, it can also be used to obtain immediately some well-known results about compact operators from C(S) to c0.

scholarly journals Hereditarily indecomposable Hausdorff continua have unique hyperspaces 2X and Cn(X)

2011 ◽  
Vol 89 (103) ◽  
pp. 49-56 ◽  
Author(s):  
Alejandro Illanes
Keyword(s):  
Hausdorff Space ◽  
Vietoris Topology ◽  
Hereditarily Indecomposable

Let X be a Hausdorff continuum (a compact connected Hausdorff space). Let 2X (respectively, Cn(X)) denote the hyperspace of nonempty closed subsets of X (respectively, nonempty closed subsets of X with at most n components), with the Vietoris topology. We prove that if X is hereditarily indecomposable, Y is a Hausdorff continuum and 2X (respectively Cn(X)) is homeomorphic to 2Y (respectively, Cn(Y )), then X is homeomorphic to Y.

Zero Marking of Past Tense in Child African American English

2014 ◽  
Vol 21 (4) ◽  
pp. 173-181 ◽  
Author(s):  
Ryan Lee ◽  
Janna B. Oetting
Keyword(s):  
African American ◽  
American English ◽  
African American English ◽  
Current Paper ◽  
Past Tense ◽  
Typically Developing ◽  
Grammatical Function

Zero marking of the simple past is often listed as a common feature of child African American English (AAE). In the current paper, we review the literature and present new data to help clinicians better understand zero marking of the simple past in child AAE. Specifically, we provide information to support the following statements: (a) By six years of age, the simple past is infrequently zero marked by typically developing AAE-speaking children; (b) There are important differences between the simple past and participle morphemes that affect AAE-speaking children's marking options; and (c) In addition to a verb's grammatical function, its phonetic properties help determine whether an AAE-speaking child will produce a zero marked form.

Ethical Issues Relevant to the Assessment of Suicide Risk in Nonclinical Research Settings

Crisis ◽  
2012 ◽  
Vol 33 (1) ◽  
pp. 54-59 ◽  
Author(s):  
Carolyn M. Wilson ◽  
Bruce K. Christensen
Keyword(s):  
Undergraduate Students ◽  
Suicide Risk ◽  
Ethical Issues ◽  
Current Paper ◽  
Self Report ◽  
Ethical Guidelines ◽  
Schizotypal Personality ◽  
Increased Risk ◽  
Genetic Features ◽  
Conducting Research

Background: Our laboratory recently confronted this issue while conducting research with undergraduate students at the University of Waterloo (UW). Although our main objective was to examine cognitive and genetic features of individuals with schizotypal personality disorder (SPD), the study protocol also entailed the completion of various self-report measures to identify participants deemed at increased risk for suicide. Aims and Methods: This paper seeks to review and discuss the relevant ethical guidelines and legislation that bear upon a psychologist’s obligation to further assess and intervene when research participants reveal that they are at increased risk for suicide. Results and Conclusions: In the current paper we argue that psychologists are ethically impelled to assess and appropriately intervene in cases of suicide risk, even when such risk is revealed within a research context. We also discuss how any such obligation may potentially be modulated by the research participant’s expectations of the role of a psychologist, within such a context. Although the focus of the current paper is on the ethical obligations of psychologists, specifically those practicing within Canada, the relevance of this paper extends to all regulated health professionals conducting research in nonclinical settings.

The Persnickety Pervasiveness of Rating Enhancement in Personality Assessment

2020 ◽  
pp. 1-13
Author(s):  
Alicia A. Stachowski ◽  
John T. Kulas
Keyword(s):  
Response Bias ◽  
Personality Assessment ◽  
Current Paper ◽  
Psychological Measurement ◽  
Shared Sets ◽  
Self Reports ◽  
Original Target ◽  
Target Profile ◽  
Do So ◽  
Normative Influences

Abstract. The current paper explores whether self and observer reports of personality are properly viewed through a contrasting lens (as opposed to a more consonant framework). Specifically, we challenge the assumption that self-reports are more susceptible to certain forms of response bias than are informant reports. We do so by examining whether selves and observers are similarly or differently drawn to socially desirable and/or normative influences in personality assessment. Targets rated their own personalities and recommended another person to also do so along shared sets of items diversely contaminated with socially desirable content. The recommended informant then invited a third individual to additionally make ratings of the original target. Profile correlations, analysis of variances (ANOVAs), and simple patterns of agreement/disagreement consistently converged on a strong normative effect paralleling item desirability, with all three rater types exhibiting a tendency to reject socially undesirable descriptors while also endorsing desirable indicators. These tendencies were, in fact, more prominent for informants than they were for self-raters. In their entirety, our results provide a note of caution regarding the strategy of using non-self informants as a comforting comparative benchmark within psychological measurement applications.

Attachment Anxiety Mitigates the Well-Being Costs of Object Attachment

2021 ◽  
Vol 42 (1) ◽  
pp. 41-56
Author(s):  
Lucas A. Keefer ◽  
Zachary K. Rothschild
Keyword(s):  
Past Research ◽  
Well Being ◽  
Attachment Anxiety ◽  
Current Paper ◽  
Personality Research ◽  
Psychological Correlates ◽  
State Study ◽  
Object Attachment ◽  
Diary Methods ◽  
State Variation

Abstract. Clinical and personality research consistently demonstrates that people can form unhealthy and problematic attachments to material possessions. To better understand this tendency, the current paper extends past research demonstrating that anxieties about other people motivate these attachments. These findings suggest that although object attachment generally correlates with poorer well-being, it may attenuate well-being deficits associated with insecurity about close relationships. The current paper presents two studies using converging correlational ( N = 394) and diary methods ( N = 413) to test whether object attachments’ association with poorer well-being is moderated by relationship uncertainties. We find that both trait (Study 1) and state (Study 2) insecurities about others eliminated, and in some cases reversed, the negative psychological correlates of object attachment. These effects, however, were only observed when focusing on between-person variation in both studies; within-person analysis demonstrated that state variation in object attachment predicted better psychological well-being. These results highlight a need for more nuanced studies of object attachment and well-being.

Sign in / Sign up

Export Citation Format

Share Document