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Can someone please recommend me an introductory topology textbook written with the functional analysis student in mind? So a book that covers the topology prerequisites a functional analysis student ought to know.

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    $\begingroup$ Many book on functional analysis contain an introductive chapter on topology. If you are interested in functional analysis I would recommend reading something like this. Reading a topology book is very interesting, but could easily lead you far astray (down the rabbit hole). $\endgroup$ – Giuseppe Negro Mar 22 at 18:11
  • $\begingroup$ Folland's Real Analysis has a chapter on point-set topology if that is what you're looking for. I believe the first part of Munkres' Topology is enough for any analyst, though it isn't with functional analysis in mind; It covers topologies and continuity, compactness and connectedness, countability and separation, metrization and paracompactness, completeness and Baire spaces. It is very accessible as well. The stress on topology in functional analysis is very overstated, inasmuch as you only need to cover the basics of set theory for mathematics in general. $\endgroup$ – I was suspended for talking Mar 22 at 18:14
  • $\begingroup$ How about Simmons' Introduction to Topology and Modern Analysis? $\endgroup$ – Lord Shark the Unknown Mar 22 at 18:28
  • $\begingroup$ Perhaps Albert Wilansky, Topology for Analysis? It's a Dover reprint (2008, of a 1970 original), so it won't break the bank! I've only dipped into it, but it looks good, both as a textbook and as a reference. Indeed, it begins: "This book is intended to serve the needs both of the beginning student and of the mature mathematician." $\endgroup$ – Calum Gilhooley Mar 22 at 18:56

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