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For 'low level', I mean those are more familiar and intuitive to me, such as integers should be much more 'low level' than the general definition of rings, or groups.

It is so often that I do not know why some concepts are important when I am studying some 'high level' mathematics (e.g. In topology: separable? second countable? Why are mathematicians studying them? Will they show up repeatedly in the world of mathematics? What are some interesting facts about them?).

Most books surely give examples, but many of them are not familiar, not 'low level' enough to me.

Let say what kind of book is considered to be 'good' for me. I think it is good if a algebra book start with some elementary number theory and then introduce the concept of group/ring (as my lecturer did), because elementary number theory is so familiar and intuitive to me (to everyone also, I think), that will make me surprised, extremely motivated and wonder:'wow, I didn't know we can consider these familiar results in such a new and powerful point of view!'. But unfortunately, those kind of book is not easy to find.

Could you please name some resources(books, papers, etc.) that introduce 'high level' concepts with 'low level' examples/facts?

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  • $\begingroup$ Based on the tags, can I take it you're looking specifically for topology resources? $\endgroup$ – Will R Sep 10 '16 at 9:10
  • $\begingroup$ Yes, but if you know some resources for other fields of mathematics, surely it is welcome. $\endgroup$ – Longitude Sep 10 '16 at 9:11
  • $\begingroup$ I don't know much in the way of topology resources, especially in what you're asking. My understanding is that general topology had kind-of a strange birth in the 20th century where people realized "hey, we can generalize the notion of continuity this way..." and as a partial consequence the notion of a topological space is really a little too general for most purposes (don't quote me on this). So I don't know how to help, sorry. $\endgroup$ – Will R Sep 10 '16 at 9:20
  • $\begingroup$ Topology mostly comes up in functional analysis or other 'high level fields'. There you won't find some low level examples. In fact, most definitions in topology usually instantly give you access to results which wont work in general topologies. $\endgroup$ – Paul K Sep 10 '16 at 9:29
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Have you seen this before: https://en.m.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes

It is a proof that there are infinite primes ("low level" result) using "topology".

The reason I put quotes around "topology" is that upon deeper inspection the proof is written in topological language but there is nothing much topological about it.

Hope you find it interesting though. I certainly was amazed by it.

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This may or may not be what you're after, but take a look at Elementary Mathematics From An Advanced Standpoint by Felix Klein. Also: Stories About Sets by Vilenkin.

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