For a given topological space (X, =), there is a coarser topology on X which is called the semi-regular topology on X (generated by regularly open subsets) and it is denoted by =δ . In this paper, we study the continuity of the group operation and the inversion mapping (ς 7􀀀 ! ς − 1) as regards the semi-regular topology =δ (not necessarily with the given topology). Then we study the said mappings with the blend of the given topology = and the semi-regular topology =δ . In the twilight of this note, we pose some questions which are noteworthy.

We generalize a theorem due to Jarosz, by proving that every almost $n$-multiplicative linear functional on Banach algebra $A$ is automatically continuous. The relation between almost multiplicative and almost $n$-multiplicative linear functional on Banach algebra $A$ is also investigated. Additionally, for commutative Banach algebra $A$, we prove that every almost Jordan homomorphism $\varphi:A\longrightarrow \mathbb{C}$ is an almost $n$-Jordan homomorphism.

Some techniques which were already used to derive automatic continuity results are chosen, they are modified, and extended results as well as generalized results are obtained. A technique of using the open mapping theorem and a technique of using the Hahn Banach extension theorem are explained. Results in connection with measurable cardinals are also obtained. Results for multiplicative linear functionals, positive linear functionals and uniqueness of topology are obtained. For example, sequential continuity of real multiplicative linear functionals on sequentially complete LMC algebras is obtained, when Michael's open problem is concerned only with boundedness of multiplicative linear functionals. The continuity of positive linear functionals on F-algebras with identity elements and involution is derived, when these functionals are continuous on the set of all involution-symmetric elements. Possibilities of extending the concept of positive linear functionals are considered to derive results for the continuity of such functionals on topological groups and topological vector spaces with additional structures. The technique for the Carpenter's uniqueness theorem is modified to derive boundedness of some homomorphisms. The entire article is oriented towards Michael's problem.

In this paper, we establish further improvements  of the Young inequality and its reverse. Then, we assert operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if $A,B\in {\mathbb B}({\mathscr H})$ are two invertible positive operators such that $0\begin{align*}& \Phi ^{2} \bigg(A \nabla _{\nu} B+ rMm \left( A^{-1}+A^{-1} \sharp_{\mu} B^{-1} -2 \left(A^{-1} \sharp_{\frac{\mu}{2}} B^{-1} \right)\right)\\& \qquad +\left(\frac{\nu}{\mu} \right) Mm \bigg(A^{-1}\nabla_{\mu} B^{-1} -A^{-1} \sharp_{\mu} B^{-1}\bigg)\bigg) \\& \quad \leq \left( \frac{K(h)}{ K\left( \sqrt{{h^{'}}^{\mu}},2 \right)^{r^{'}}} \right) ^{2} \Phi^{2} (A \sharp_{\nu} B),\end{align*}where $r=\min\{\nu,1-\nu\}$, $K(h)=\frac{(1+h)^{2}}{4h}$,  $h=\frac{M}{m}$, $h^{'}=\frac{M^{'}}{m^{'}}$ and $r^{'}=\min\{2r,1-2r\}$. The results of this paper generalize the results of recent years.

In this work the notion of a bornological linearly topologized module over a discrete valuation ring is introduced and it is shown that certain semimetrizable linearly topologized modules are bornological. The main result is a characterization of bornological linearly topologized modules, from which the completeness and the quasi-completeness of certain linearly topologized modules of continuous linear mappings are derived.

Let A and U be Banach algebras, θ,be a nonzero character on A and let A ×θ,U be the corresponding Lau product Banach algebra. In this paper we investigate derivations and multipliers of A ×θ,U and study the automatic continuity of these maps. We also study continuity of the derivations for some special cases of U and the Banach (A ×θ,U)-bimodule X and establish various results in this respect. Some of the results are devoted to find conditions under which one can represent a derivation on A ×θ,U as sum of two derivations in such a way that one of them is continuous. Some examples are also given.

In this paper, a new definition of bounded fuzzy linear order homomorphism on $I$-topological vector spaces is introduced. This definition differs from the definition of Fang [The continuity of fuzzy linear order-homomorphism. J. Fuzzy Math. {bf 5}textbf{(4)}(1997), 829--838]. We show that the “boundedness" and “boundedness on each layer" of fuzzy linear order homomorphisms do not imply each other. On the basis, characterizations of continuity of fuzzy linear order-homomorphisms, and the relation between continuity and boundedness are studied.

Let X and Y be vector spaces. We show that a mapping f : X ®Y satisfies the functional equation, f  (x1 +S2dj=2(-1)j xj)-f (x1+S2dj=2(-1)j-1xj)=2S2dj=2 (-1)j (xj)if and only if the mapping f : X ® Y is Cauchy additive, and prove the Cauchy-Rassias stability of the above functional equation in Banach modules over a unital C*-algebra, and in Poisson Banach modules over a unital Poisson C*-algebra. Let A and B be unital C*-algebras, Poisson C*-algebras or Poisson JC*-algebras. As an application, we show that every almost homomorphism h  A®B of A into B is a homomorphism when h(2nuy) = h(2nu)h(y) or h(2nu°y) = h(2nu) o h(y), for all unitaries u Î A, all y Î A, and n = 0, 1, 2, Moreover, we prove the Cauchy-Rassias stability of homomorphisms in C*-algebras, Poisson C*-algebras or Poisson JC*-algebras.