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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?2012-10-30
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    @RobertIsrael The words "closed","open""compact" seem to be absent.2012-10-30
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    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*.)2012-10-31
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    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.2013-01-25
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    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.2013-01-27

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