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I have noticed that certain texts like Terry Tao's Analysis I, Bartle and Sherbert's Real analysis as well as Michael Spivak do not use the language of topology in their exposition.

A text like Walter Rudin's Principles of Mathematical Analysis has a separate chapter on topology.

This has confused me.As someone who will shortly study real analysis, which approach should be the preferred one?

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I don't know those texts, but you really can't go very far in analysis without some point set topology. "Open", "closed", "compact", and even "continuous" and "converge" are part of the language of topology. Do they not use those words? – Robert Israel Oct 30 '12 at 6:22
@RobertIsrael The words "closed","open""compact" seem to be absent. – user43081 Oct 30 '12 at 6:37
As I recall, Bartle & Sherbert is a sadly watered-down version of Bartle’s excellent The Elements of Real Analysis, which does quite a bit of topology. (It even defines compactness correctly, in terms of open covers. So, come to think of it, does Gaughan’s nice little Introduction to Analysis.) – Brian M. Scott Oct 31 '12 at 1:35
If you are referring to Michael Spivak's Calculus, that is not intended to be an analysis textbook, though its content does overlap significantly with what many undergraduate analysis classes would typically cover. – Gyu Eun Lee Jan 25 '13 at 21:17
I have never looked at Bartle-Sherbert, but I can attest that Terry Tao's book and Spivak's book use the language of point-set topology, at least in $\mathbb{R}^n$. Look harder. – Adam Smith Jan 27 '13 at 21:30

Historically speaking, perhaps an analysis course should begin with Fourier's approach to problems regarding the conduction of heat. (For more on this, see e.g. here.)

Of course, many math curricula are organized and re-organized (and re-re-organized) in ways differing radically from their material's original development. For example, Galois Theory is often offered as a sort of "Abstract Algebra II" course, though its creation (e.g. work by Galois himself) did not rely on the group theoretic arguments that you would now likely learn first in an "Abstract Algebra I" class.

As a matter of personal preference, I think it's helpful to start building a topological vocabulary in your first real analysis course (e.g. words such as those mentioned above by Robert Israel). One could argue that learning these definitions in a specific context will be confusing when moving to a more general context (e.g. what compact means in a real analysis course vs. what it will mean in a topology course). Others might argue that having specific examples will help when you go on to study objects of greater generality.

There are a lot of topics that could be covered in an introductory analysis course, and it's quite possible that what is deemed important enough for inclusion is related mostly to the instructor's background (e.g. someone with a background in harmonic analysis might take a more Fourieresque approach in lieu of introducing concepts from point set topology).

I cannot tell you what "should" be done, but I would include a portion on the standard topology in Euclidean space in my own course on analysis, partly because I think the terminology should be seen sooner rather than later, and partly because I find the material particularly interesting and fun to play around with.

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no , you lose a general metric space perspective such as Rudin shows. Most of the better real analysis books include some topology.

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It really depends on what you consider an analysis course at the undergraduate level. If you come from a strong background in calculus using a theoretical approach like Spivak's Calculus or McCleur's Honors Calculus, where a careful treatment of limits is given, then an introduction to point set topology on Euclidean spaces-such as Rudin or Charles Chapman Pugh's Real Mathematical Analysis (my personal favorite)- is appropriate for a first analysis course. If you have a pencil pushing, watered down calculus training where you basically memorize trigonometric identities and learn how to differentiate and integrate, then a point set topology level course is going to be very tough sledding and a "baby analysis" course a la Kenneth Ross'Elementary Analysis:The Theory Of Calculus. So you really need to evaluate your background before deciding.

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