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I have a plan to self-study real analysis,but I'm not sure about myself I complete read Calculus,linear Algebra ,Number theory. But I saw in some thai textbook mention about topology.Is it important for real analysis ? I have more question . Is calculus by spivak will help for studying real analysis?

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    $\begingroup$ They are related subjects of study, but neither is dependent on the other. Knowing some topology will help you digest some topics better, while it is also true that knowing some analysis can help you when learning topology $\endgroup$
    – Asinomás
    Oct 16, 2016 at 18:43
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    $\begingroup$ Most introductory real analysis textbooks will go over any topology that they require (connectedness, continuity in $\Bbb R$, etc). $\endgroup$
    – user137731
    Oct 16, 2016 at 18:44
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    $\begingroup$ Typically at universities, it's just multivariable calculus that's a prereq to analysis. It's actually more likely the case that you'll see analysis listed as a prereq for a first course in topology. Any topology you end up doing in a first analysis course will start with the basics, so there's no need to trouble yourself with that. Spivak's book is quite good, but I'd also recommend "Analysis with an Introduction to Proof" by Steven R Lay. Best of luck! $\endgroup$
    – AlkaKadri
    Oct 16, 2016 at 19:05
  • $\begingroup$ See also this MSE post: Good textbooks for Analysis and Topology $\endgroup$
    – amWhy
    Oct 16, 2016 at 19:05

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From the Texas A&M University catalog, this is the description of the course MATH 409, a first course in advanced calculus. This is a bridge to the real analysis sequence.

Axioms of the real number system; point set theory of $\mathbb{R}^1$; compactness, completeness and connectedness; continuity and uniform continuity; sequences, series; theory of Riemann integration.

While "compactness" appears in the description, the texts used for this course don't mention topology. Topology does help.

I'll show the descriptions for other courses in real analysis. First, a senior-level bridge to graduate analysis, MATH 446:

Construction of the real and complex numbers; topology of metric spaces, compactness and connectedness; Cauchy sequences, completeness and the Baire Category Theorem; Continuous Mappings; introduction to Point-Set Topology.

The topology of metric spaces is used a lot in that course. Next is its successor, MATH 447:

Riemann-Stieltjes integration; sequences and series of functions; the Stone-Weierstrass and Arzela-Ascoli Theorems; introduction to Lebesgue measure theory and integration.

Here is where some more topology helps. We talk about Borel sets, sigma-algebras, and talk about measurability in terms of open sets. Finally is the first graduate course in real analysis, MATH 607:

Lebesgue measure and integration theory, differentiation, Lp-spaces, abstract integration, signed measures; Radon-Nikodym theorem, Riesz representation theorem, integration on product spaces.

The topology can help here as well. In short, a good prerequisite for studying real analysis is topology, maybe some basic probability theory (at the undergraduate level), and linear algebra. If you haven't seen the topics at the equivalent of MATH 409 from above, I suggest you start there. Spivak is good, but I find that Understanding Analysis by Stephen Abbott is a really good starter for advanced calculus.

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