Basic concepts of topology

Introduction

The notion of a set, while indicating that certain objects —the elements that comprise it— have something in common, does not give us any sense of the closeness between these elements. On the other hand, if we consider, for instance, the real numbers, this notion of closeness is evident. We know, for example, that the number 2 is much closer to 1 than 423 is. The concept of a topology on a set, which we will define below, aims to precisely capture this notion of closeness, which, as we will see, allows for many gradations.

Un espacio topológico consiste en un par $(X, \mathcal{T})$, donde $X$ es un conjunto y $\mathcal{T}$ es una colección de subconjuntos de $X$ satisfaciendo las siguientes condiciones:

Second subtopic