Usually a calculus or real analysis book has the following topics
I am searching a book with those topics but such that each topic is developed in the most abstract way, i.e. convergence and continuity maybe from the perspective of topology or metric spaces and integrability from Lebesgue perspective. Thank you so much for your help.
On
I'm not sure exactly what your purpose is, but you could perhaps have a look at Handbook of Analysis and Its Foundations by Eric Schechter.
(See also the description on the author's own web page.)
On
See what you think of Gleason, Fundamentals of Abstract Analysis, https://www.maa.org/press/maa-reviews/fundamentals-of-abstract-analysis
There's also Loomis and Sternberg, Advanced Calculus, https://www.maa.org/press/maa-reviews/fundamentals-of-abstract-analysis
Table of contents:
Introduction
Vector Spaces
Finite-Dimensional Vector Spaces
The Differential Calculus
Compactness and Completeness
Scalar Product Spaces
Differential Equations
Multilinear Functionals
Integration
Differentiable Manifolds
The Integral Calculus on Manifolds
Exterior Calculus
Potential Theory in $E^n$
Classical Mechanics
Do you read French or, say, Russian? Analysis (Cours d'analyse) by Laurent Schwartz is pretty close to your request. But I do not think it was translated into English.
Another book you may try is the recent book by Barry Simon "Real Analysis: A Comprehensive Course in Analysis, Part 1" (Amazon link)
The major difference is that the latter book does assume a good background (it is listed in introduction), whereas the former starts from scratch and self contained (personal remark: I would kill myself if it was my first book in analysis).