Resources on Mathematics for the Educated non-Mathematician

I was wondering what the best books are for giving an educated non-mathematician the gist of real mathematics. Not to say that I am by any means a real mathematician, but it is one of my goals to pursue graduate school and further studies in mathematics.

My father is educated in medicine and in physics, chemistry, and biology- primarily the latter two. He is the type of person with extremely meticulous study habits, and it seems as though he wants to understand my mathematical interests. My primary interests are topology/analysis, and I was wondering if someone could give me a good recommendation of literature I could give him so that he could understand the goals of modern mathematics.

I'm not looking for in-depth, or technical books on topology/analysis, I own those already- I just want a book that can put the field in perspective in a somewhat conversational, but not oversimplified manner$^\dagger$.

$\dagger$ Let me just emphasize that I am looking for books catered to people who are specialists in other fields, but didn't study mathematics rigorously outside of their own field.

• Terence Tao's Analysis series is good for introductory reading. – Kushal Bhuyan Dec 5 '15 at 7:59
• For topology a book that I love is Topology without tears of Morris. The Knuth's books generally are very fine too. I think a good book of topology, another of set theory, analysis and algebra are the core of maths. – Masacroso Dec 5 '15 at 8:14