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Weakly semi-open functions between topological spaces are defined as dual to the weak semi-continuous. Connections between this function and other existing topological notions are established and separate examples are given. It is shown that weakly semi-open functions on regular spaces are semi-open. We also introduce and study the concept of weakly semi-closed functions.
We introduce a new notion of weakly $M$-open functions as functions defined between sets satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. The functions enable us to formulate a unified theory of weak openness , weak semi-openness , weak preopenness , and weak $\beta$-openness .
These results stem from a course on ring theory. Quantum planes are rings in two variables $x$ and $y$ such that $yx=qxy$ where $q$ is a nonzero constant. When $q=1$, a quantum plane is simply a commutative polynomial ring in two variables. Otherwise, a quantum plane is a noncommutative ring.
Our main interest is in quadratic forms belonging to a quantum plane. We provide necessary and sufficient conditions for quadratic forms to be irreducible. We find prime quadratic forms and consider more general polynomials. Every prime polynomial is irreducible and either central or a scalar multiple of $x$ or of $y$. Thus, there can only be primes of degree 2 or more when $q$ is a root of unity.
We discuss some general methods for structuring undergraduate research projects. As an example, we follow a project which occurred between the two authors who at one time had the student-advisor relationship. We discuss realistic goals of undergraduate research, reflect on the methods and outcomes of our project, and suggest ideas for future work with undergraduates.
In this work, it is shown that the group of isometries of the plane with respect to the Chinese Checkers metric is the semi-direct product of the Dihedral group $D_8$ and $T(2)$, where $D_8$ is the (Euclidean) symmetry group of the regular octagon and $T(2)$ is the group of all translations of the plane. Furthermore, some properties of the CC-plane are studied and the area formula for a triangle is given.
In this paper, we characterize the rings whose prime spectrum satisfy some topological notions related to the lower separation axioms. In order to do so, the notions of $\gamma$-ring, $\varepsilon$-ring, and $ES$-ring are introduced.