We recall that, in a topological space, a point is said to be an interior point of a subset if the latter is a neighborhood of the point in question. The set of interior points of a subset is called the interior of this subset. It is well-known that a subset is open if and … Lire la suite de A characterization of the existence of an interior point
Open set
An exercise on a characterization of the existence of an interior point
We recall that, in a topological space, a point is said to be an interior point of a subset if the latter is a neighborhood of the point in question. The set of interior points of a subset is called the interior of this subset. It is well-known that a subset is open if and … Lire la suite de An exercise on a characterization of the existence of an interior point
Topologies from interior operators
As seen in a preceding entry of this blog, topologies on a set can be defined by the means of closure operators. In this note, we prove that topologies can also be constructed using interior operators. In fact, the note is a solution of an exercise formulated previously. We repeat the statement of the exercise … Lire la suite de Topologies from interior operators
An exercise on topologies from interior operators
In a preceding entry of this blog, we saw that topologies arise from closure operators. This note raises the question of whether this is also valid for interior operators. The question is framed precisely in the exercise below. To approach the issue with serenity, it should be remembered here that closure and interior are dual … Lire la suite de An exercise on topologies from interior operators
Properties of the frontier operator
The framework of this publication is an exercise of the volume on General Topology by Bourbaki (cf. Exercise 5 of the section §1 in Chapter I). The paper, divided into five sections, is devoted to the frontier operator and some of its properties. The first section gives the definition of the frontier and some elementary … Lire la suite de Properties of the frontier operator
Formalis Mathematica 