How should one use a book like this? Is it to get an overview of a topic before diving in? I don’t think I’ve ever learnt any mathematics from reference works, so I’m curious as to their intended audience.
“A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n)”
At this point, either you know what all of those words mean or you don’t. If you do, great! You’re done. If not, you either keep digging deeper into the various terms or you start seriously considering reading one or more of the curated reference books listed at the end of each entry.
Over time you develop the “mathematical maturity” that you don’t need to do a deep dive into the books and can mostly just use the reference.
I'm not sure. I only have a rather rudimentary understanding of topology, so I do understand the definition of a manifold on a technical level, but I don't know any interesting examples or theorems about them so it wouldn't be immediately clear to me why something being a submanifold is worth mentioning.
Similarly, I don't think that just reading the definition really gives you a good understanding of groups. You probably want to work through some examples of groups, and arguably, the importance of groups doesn't really become clear until you've encountered group actions.
> Say you’re reading a paper or trying to implement some technology that uses a mathematical concept you aren’t familiar with
In such a case you're not interested in either manifold or sub-manifold or group in and of itself. So a lack of familiarity with theorems isn't an impediment.
Reading mathematical definitions on their own just doesn't give you a whole lot of context about the objects they're describing.
A way to find good ones is to look at some university webpages, to see what books they use in 1-level and 2-level classes. (Of course, start with 1-level.). Those textbooks will be more expansive, with interesting diagrams, problem sets, and so forth. And they will use fancy typesetting patterns, like insets in boxes for subtopics, etc.
I suspect quite a few purchasers will be university teachers who want to have this on their shelves, for when students come by and ask for a book to borrow overnight to brush up on a topic.
In a similar topic, if someone is considering a career in mathematics, I like the book, "A Mathematician's Survival Guide: Graduate School and Early Career Development." It applies to both pure and applied mathematicians, but it does a good job of walking through undergraduate studies all of the way to being a professor. Not all mathematicians end up in the professoriate, but the graduate school information is still valuable.