Set theory and topology. An introduction to the foundations of analysis. Part II: Topology - Fundamental notions

TL;DR

This paper introduces foundational concepts of topology, including spaces, convergence, and continuity, with formal theorems and construction methods, emphasizing their applications to real numbers in analysis.

Contribution

It provides a formal, axiomatic foundation for topology, detailing fundamental notions and construction techniques with applications to analysis.

Findings

Formal definitions of topological spaces and convergence

Methods for constructing topological spaces

Applications of topology to real analysis

Abstract

We provide a formal introduction into the classic theorems of general topology and its axiomatic foundations in set theory. In this second part we introduce the fundamental concepts of topological spaces, convergence, and continuity, as well as their applications to real numbers. Various methods to construct topological spaces are presented.