Topologies from interior operators

/ Christian Nguembou Tagne

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 in question here, as a reminder.

Exercise.

Let X be a set. If i:\mathcal{P}(X)\rightarrow\mathcal{P}(X) is an operator which carries subsets of X into subsets of X, and \mathfrak{O} the family of all subsets such that i(A)=A, under what conditions will \mathfrak{O} define a topology on X and i be the interior operator relative to this topology?

Solution of the exercise as PDF file

Please support our blog with a donation!

Choose an amount

€5,00
€15,00
€100,00

Or enter a custom amount :

€

Your contribution is appreciated.

DONATE
Publicités

Laisser un commentaire

Ce site utilise Akismet pour réduire les indésirables. En savoir plus sur la façon dont les données de vos commentaires sont traitées.