For any affine semigroup S, the set S∪{∞} has a natural structure; additionally, if S is endowed with discrete topology, then can be studied as one-point compactification of S. In this article, we study derivations on algebra C[S∪{∞}] in relation to C[S] considering metrizable topology induced by S∪{∞}.

in this paper we investigate generalized topologies generated by a subbase ofpreorder relators and consider its application in the concept of the complement. we introducethe notion of principal generalized topologies obtained from the new type of open sets andstudy some of their important properties.

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...

We study the geometric response of three-dimensional non-Hermitian crystalline systems with nontrivial point-gap topology. For fourfold rotation symmetry, we show that in presence disclination lines a total Frank angle which is an integer multiple $2\pi$, there can be one-dimensional topology along direction lines. This results disclination-induced skin effects. By doubling Hamiltonian to Hermi...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...