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I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started reading their second chapter, "Sheaves of Sets" but I can not follow. Is there a text, maybe, that someone can recommend, that introduces the basics of topology, from a categorical view point? I do not want a career in topology, I only want to learn what is needed in order to be able to follow material like what is included in the chapter mentioned above. Thanks for any help!

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    $\begingroup$ Topology springs from the desire to extend ideas and theorems from analysis of the real line and functions on it to more general spaces. The open intervals in $\Bbb{R}$ and open balls in $\Bbb{R}^n$ play a special role. Hence, I recommend that you first get comfortable with the basics of topology in Euclidean space. $\endgroup$ – Sammy Black Jun 10 '14 at 6:17
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    $\begingroup$ The chapter of Mac Lane and Moerdijk's book is quite elementary. Every thing is defined in there except for the basic concepts (topology on a set, continuous map, etc.). You shouldn't go further in the book if you're not comfortable with such basic notions. Indeed, the next chapter is about Grothendieck's topologies (and topos), that is a generalization of topologies for any (small) category. It is very useful to have topological intuition to get most of the properties of Grothendieck's topologies. $\endgroup$ – Pece Jun 10 '14 at 6:28
  • $\begingroup$ @Pece I would like to know about this "topological intuition" you mention... How does one gain it, you think, if they are not familiar with topology at all and they just want to study category theory (or topos theory)? $\endgroup$ – Bill_Werden Jun 10 '14 at 6:44
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    $\begingroup$ You can probably just skip the chapter on classical sheaf theory entirely if you have a strong grip on category theory. But any serious mathematician must know at least the definition of topological space, continuous map, etc. $\endgroup$ – Zhen Lin Jun 10 '14 at 7:13
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    $\begingroup$ @AlistairDermont The amount of background you need is minimal. Even Wikipedia would suffice. $\endgroup$ – Zhen Lin Jun 10 '14 at 13:52
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Hi AlistairDermont and welcome to Mathematics.SE.

If you wish to learn a bit of general topology and then use it for whatever legal purpose it may serve, as they say, I would suggest that you look at Schaum's General Topology by S. Lipschutz. It is perfect for self-learning with its wealth of solved exercises. Chap. 5 starts with the real juicy stuff. In parallel I also suggest you look at Wikipedia's related pages and references therein (Munkres and Kelley are standard).

This page will give you an overview of what's available:

http://en.wikipedia.org/wiki/List_of_general_topology_topics

If you wish to study basic category theory without general topology then the best option is Awodey's book whose first edition can also be found online.

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In order to learn category/topos theory, it is essential that you get comfortable with topology. For example, in order to get the intution for Grothendieck topologies, you should be familiar with topology.

This turns the question more into a reference request for topology textbooks. Let me recommend a few books which I personally appreciated. I liked Topology without Tears, S. Morris. Munkres is a great book; in order to gain intuition for Munkres, do the exercises. General Topology by Stephen Willard and Basic Topology by M.A. Armstrong are also pretty good books. I wouldn't recommend this for studying, but Singer and Thorpe, Lecture Notes on Elementary Topology and Geometry is a great book. It does not have any exercises, and so is not a textbook like Munkres. (You could check out Allen Hatcher's Notes on Introductory Point-Set Topology is you wish to.)

Lastly, check out Allen Hatcher's recommendations.

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The book by Mendelson (Introduction to topology) is quite nice.

Near the end it has a very introductory discussion of the fundamental group.

As well as the basic language of topological spaces, a very useful thing to learn would be the theory of covering spaces (which builds on the theory of the fundamental group, at least the way it is usually presented). The point is that a basic example of a sheaf of sets is the sheaf of sections of a covering space (and it's pretty hard to appreciate the concept of sheaf without being able to understand this example).

There are many texts which explain covering spaces, of course; the first chapter of Hatcher's book on algebraic topology is one such place. (Note that you don't need to read on and learn the material of the later chapters, although of course that material is also very interesting; I am specifically recommending covering space theory here. Unfortunately most general topology texts don't cover this theory.)

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