Conceptual Topology & Manifolds books

[Winzer]

I am looking for books that introduce the fundamentals
of topology or manifolds. Not looking for proofs and rigor.
Something that steps through fundamental theorems in the
field, but gives conceptual explanations.

 

[Sankaku]

This is the closest thing I can think of is "The Shape of Space":

https://www.amazon.com/dp/0824707095/?tag=pfamazon01-20

Most of the material in point-set Topology is very abstract and often non-intuitive. I doubt you will find a "conceptual" book that "steps through fundamental theorems" because you really need the proofs to get anywhere.

 

[xristy]

One intuitive book is Griffiths' Surfaces a few theorems such as the Konigsberg bridge problem and the Euler characteristic are demonstrated in simple language along with non-orientable surfaces and so on.

Crossley's Essential Topology unfortunately includes some proofs but is not big on rigor. I suspect that it will be difficult to find a book that doesn't have some proofs.

 

[homeomorphic]

You could try Prasolov's Intuitive Topology. I haven't read it, but I took a look inside and it seems like that sort of thing.

There's also a book, Algebraic Topology: An Intuitive Approach.

Armstrong: Basic Topology has an introduction along the lines you have in mind.

Hilbert and Cohn Vossen's book, Geometry and the Imagination has a chapter on topology.

Another thing you might try is to look at historical references. It should be kept in mind that the study of topology really precedes point-set topology in its modern form. One of the earliest results was the classification of surfaces by Riemann and Mobius, independently, back in the 19th century. Whereas, I think point-set sort of reached a pretty modern form in the 1920s. Another interesting thing to look at, which I haven't done yet, is to read Poincare's old papers. Also predating point-set.

 

[rezeew]

I understand the importance of having a strong conceptual understanding of a subject before delving into rigorous proofs and theorems. In the field of topology and manifolds, there are several books that can provide a conceptual introduction to these topics without overwhelming the reader with technical details.

One book that I would recommend is "Introduction to Topology: Pure and Applied" by Colin Adams and Robert Franzosa. This book presents the fundamental concepts of topology in an intuitive and accessible manner, without getting bogged down in formal proofs. It also includes many real-world examples and applications of topological concepts, making it a great resource for those looking for a conceptual understanding.

Another helpful book is "Topology: A Categorical Approach" by Bradley T. Franks. This book uses category theory to introduce the basic concepts of topology, providing a unique and conceptual approach to the subject. It also includes many visual aids and examples to aid in understanding the material.

For an introduction to manifolds, "An Introduction to Manifolds" by Loring W. Tu is a great resource. This book presents the fundamental ideas of manifolds in an intuitive and visual manner, making it accessible to those without a strong mathematical background. It also includes many exercises and examples to reinforce the concepts being taught.

Overall, these books provide a conceptual introduction to the fundamentals of topology and manifolds, without getting bogged down in rigorous proofs. They are great resources for those looking to gain a strong understanding of these topics before delving into more technical material.