# Visualization in a metric space?

Metric space is the first phase of abstract analysis,even in real analysis we do everything with proper rigor and do not depend on pictures only,although pictures are necessary to understand a rough sketch behind the proof.Similarly there is nothing like visualization because metric space is a set of 'anything' not necessarily some geometrical points,it may be a set of anything like screw,pens,pencils,humans,matrices etc.So,it is not correct to arrive at a conclusion only by drawing simple pictures because pictures may also be deceptive in certain cases(for example,discrete metric).But still we sometimes need to draw a rough picture like a sphere to represent open balls although balls do not always are like 'balls' 'round'.Should I draw a rough simplified picture to understand how to proceed with the proof or problem and then translate the intuition into proper logic by formally writing what I did in the pictures.Is it the correct approach,as it provides me with both visualization and rigour. Can someone suggest me also some good book that discusses the visual approach also of metric spaces?

• $\mathbb {R, R^2, \text{ and } R^3}$ are examples of metric spaces, and you can draw subsets of them – J. W. Tanner Feb 16 at 14:02
• @J.W.Tanner Is it safe to draw a simple diagram(beacuse I am verifying it with rigor)?And what about some reference? – Kishalay Sarkar Feb 16 at 14:15
• I think pictures are helpful for understanding, even if they don't provide rigorous proofs. I'm sorry I don't have a reference; that's why I made a comment, not a full answer – J. W. Tanner Feb 16 at 14:21
• Are you looking for a book that says "yes, some people might imagine $\mathbb R^2$ or $\mathbb R^3$ and with practice be able to use that effectively to conceive of ideas for proofs that apply to all metric spaces"? There might be such a book, but I don't think you need a book to tell you that. If it's working for you, isn't that enough? – Mark S. Feb 16 at 18:31
• @MarkS. I am not looking for a book to say yes.I want a book that excercises the visual approach. – Kishalay Sarkar Feb 17 at 1:50