In this paper, we prove the following statements: \item {(1)} There exists a Hausdorff Lindelöf space $X$ such that the Alexandroff duplicate $A(X)$ of $X$ is not discretely absolutely star-Lindelöf. \item {(2)} If $X$ is a regular Lindelöf space, then $A(X)$ is discretely absolutely star-Lindelöf. \item {(3)} If $X$ is a normal discretely star-Lindelöf space with $e(X)< \omega _1$, then $A(X)$ is discretely absolutely star-Lindelöf.
We show that an effect tribe of fuzzy sets T⊆[0,1]X with the property that every f∈T is B0(T)-measurable, where B0(T) is the family of subsets of X whose characteristic functions are central elements in T, is a tribe. Moreover, a monotone σ-complete effect algebra with RDP with a Loomis-Sikorski representation (X,T,h), where the tribe T has the property that every f∈T is B0(T)-measurable, is a σ-MV-algebra.
The restrained domination number $\gamma ^r (G)$ and the total restrained domination number $\gamma ^r_t (G)$ of a graph $G$ were introduced recently by various authors as certain variants of the domination number $\gamma (G)$ of $(G)$. A well-known numerical invariant of a graph is the domatic number $d (G)$ which is in a certain way related (and may be called dual) to $\gamma (G)$. The paper tries to define analogous concepts also for the restrained domination and the total restrained domination and discusses the sense of such new definitions.
Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1, d_2, \dots , d_n)$, where $d_1 \ge d_2 \ge \dots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho (G)$ and $\mu (G)$, respectively. We determine the graphs with \[ \rho (G) = \frac{d_n - 1}{2} + \sqrt{2m - nd_n + \frac{(d_n +1)^2}{4}} \] and the graphs with $d_n\ge 1$ and \[ \mu (G) = d_n + \frac{1}{2} + \sqrt {\sum _{i=1}^n d_i (d_i-d_n) + \Bigl (d_n - \frac{1}{2} \Bigr )^2}. \] We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.
In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming $2^{\aleph _0}=2^{\aleph _1}$, there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.
Let P be a topological property. A space X is said to be star P if whenever U is an open cover of X, there exists a subspace A ⊆ X with property P such that X = St(A,U), where St(A, U) = ∪ {U ∈ U : U ∩A ≠ ∅}. In this paper, we study the relationships of star P properties for P ∈ {Lindelöf, compact, countably compact} in pseudocompact spaces by giving some examples.
In this paper we investigate the relationship between the statistical (or generally I-convergence) of a series and the usual convergence of its subseries. We also give a counterexample which shows that Theorem 1 of the paper by B. C. Tripathy ''On statistically convergent series'', Punjab. Univ. J. Math. 32 (1999), 1–8, is not correct.