In this paper examples are given to show that s-regular and s-normal are independent; that s-normal, and s-regular are not semi topological properties; and that (S(X),E(X)) need not be I even if (X,T) is compact, s-normal, s-regular, semi-T2, and T0. Also, it is shown that for each space (X,T), (S(X) ,E(X)), (S(XO) ,E(Xo)), and (S(Xs0) ,E(Xs0)) are homeomorphic, where (X0, Q(X0)) is the T0-iden...

In this paper, the gradual real numbers are considered and the notion of the gradual normed linear space is given. Also some topological properties of such spaces are studied, and it is shown that the gradual normed linear space is a locally convex space, in classical sense. So the results in locally convex spaces can be translated in gradual normed linear spaces. Finally, we give an examp...

1. Provocative example 2. Natural function spaces on the circle S = R/2πZ 3. Topology on C∞(S1) 4. Pointwise convergence of Fourier series 5. Distributions: generalized functions 6. Invariant integration, periodicization 7. Hilbert space theory of Fourier series 8. Levi-Sobolev inequality, Levi-Sobolev imbedding 9. C∞(S1) = limC(S) = limH(S) 10. Distributions, generalized functions, again 11. T...

In this paper, the concept of somewhat-connected space will be introduced and characterized. Its connection with the other well-known concepts such as the classical connectedness, the $omega_theta$-connectedness, and the $omega$-connectedness will be determined. Moreover, the concept of textit{somewhat}-continuous function from an arbitrary topological space into the product space will be chara...

In this article, we define the vector valued multiple of $chi^{2}$ over $p$-metric sequence spaces defined by Musielak and study some of their topological properties and some inclusion results.

1. A confusing example 2. Natural function spaces 3. Topology on limits C∞(S1) 4. Distributions (generalized functions) 5. Invariant integration, periodicization 6. Hilbert space theory of Fourier series 7. Completeness in L(S) 8. Sobolev lemma, Sobolev imbedding 9. C∞(S1) = limC(S) = limHs(S) 10. Distributions, generalized functions, again 11. The confusing example explained 12. Appendix: prod...

in this paper, vector ultrametric spaces are introduced and a fixed point theorem is given forcorrespondences. our main result generalizes a known theorem in ordinary ultrametric spaces.