A Guide to Topology
This book is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.
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accumulation point basis boundary calculus called Cantor set Cauchy certainly closed sets closure collection compact set compact-open topology compactification complement connected contains continuous function critical point define dense dimension disc disjoint open sets distinct points elements embedding empty set equicontinuous equivalent Euclidean space Example family of functions Figure finite subcovering Hausdorff space height function hence Hilbert cube homeomorphism idea infinite integers interior inverse image Krantz Lebesgue number lemma level sets Mathematical metric space Morse theory noncut point open ball open cover open interval open sets paracompact partition of unity plane point x e pointwise convergence polynomial Proof Proposition pseudometric quotient topology rational numbers real line real numbers Ross Honsberger second countable Section separation axiom sequence space A space subbasis Suppose topological space topology of pointwise torus Tychanoff uniform convergence uniform space unit interval usual topology vector space Weierstrass Approximation Theorem