On a Connected $T_{1/2}$ Alexandroff Topology and $^*g\hat{\alpha}$-Closed Sets in Digital Plane

The Khalimsky topology plays a significant role in the digital image processing. In this paper we define a topology $\kappa_1$ on the set of integers generated by the triplets of the form $\{2n, 2n+1, 2n+3\}$. We show that in this space $(\mathbb{Z}, \kappa_1)$, every point has a smallest neighborhood and hence this is an Alexandroff space. This topology is homeomorphic to Khalimskt topology. We prove, among others, that this space is connected and $T_{3/4}$. Moreover, we introduce the concept of $^*g\hat{\alpha}$-closed sets in a topological space and characterize it using $^*g\alpha o$-kernel and closure. We investigate the properties of $^*g\hat{\alpha}$-closed sets in digital plane. The family of all $^*g\hat{\alpha}$-open sets of $(\mathbb{Z}^2, \kappa^2)$, forms an alternative topology of $\mathbb{Z}^2$. We prove that this plane $(\mathbb{Z}^2, ^*g\hat{\alpha}O)$ is $T_{1/2}$. It is well known that the digital plane $(\mathbb{Z}^2, \kappa^2)$ is not $T_{1/2}$, even if $(\mathbb{Z}, \kappa)$ is $T_{1/2}$.