An open (resp., closed) subset A of a topological space (X, T ) is called F-open F-closed) set if cl(A)\A A\int(A)) finite set. In this work, we study the main properties these definitions and examine relationships between F-closed sets with other kinds such as regularly open, closed, sets. Then, establish some operators F-interior, F-closure, F-derived...etc., using At end introduce F-continuo...

We investigate the possibility of crossing a phantom divide line in extension symmetric teleparallel gravity or $f(Q)$ gravity, where $Q$ is non-metricity. study cosmic evolution effective equation state parameter for dark energy considering exponential, logarithmic, and combined theories. Moreover, exponential model behaves like $\Lambda$CDM at high redshifts before deviating to $\omega_{eff}<...

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.