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I would like to know if anyone could suggest a general topology book for a deeper understanding of probability at advanced level. If there is an advanced topology book oriented to probabilists, I would quite appreciate it.

I have made the same question at here, but it did not get an answer so far.

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    $\begingroup$ Why do you think probability theory requires much general topology? Do you have something specific in mind? $\endgroup$
    – Michael Greinecker
    Feb 8, 2022 at 0:54
  • $\begingroup$ @MichaelGreinecker I would like to be prepared to study "Real Analysis and Probability" from Dudley with a solid background in topology so that I could complement my background on the subject before start reading it. $\endgroup$
    – user1234
    Feb 8, 2022 at 0:58
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    $\begingroup$ Dudley introduces general topology mostly because it is also a general-purpose real analysis book. He even has an appendix E in which he explains why probability theory nowadays mostly works with Polish spaces (separable completely metrizable spaces) that avoid most of the complications and subtleties of general topology, and not the setting of locally compact spaces that used to be quite popular at the height of Bourbaki. $\endgroup$
    – Michael Greinecker
    Feb 8, 2022 at 7:17
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    $\begingroup$ "I have made the same question at here, but it did not get an answer so far" (i.e. in the first 8 hours after you asked it) - it might help if you respond to the comment there trying to clarify what it is you're looking for. Otherwise no one is likely to answer.... $\endgroup$
    – James Martin
    Feb 8, 2022 at 16:03
  • $\begingroup$ Bogachev's book "Measure Theory" has a quite strong topological flavour. Maybe it will be helpful probability oriented students. $\endgroup$
    – Taras Banakh
    Feb 12, 2022 at 0:25

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Chapter 4 on general topology in Folland's real analysis book [1] should suffice. As noted in the comments, the setting of Polish spaces is more relevant for Modern probability. The lovely classic [2] is wonderful to sample the interactions of measure and topology, you can just read the first ten chapters (pages 1-44).

[1] Folland, Gerald B. Real analysis: modern techniques and their applications. Vol. 40. John Wiley & Sons, 1999. https://www.amazon.com/Real-Analysis-Modern-Techniques-Applications/dp/0471317160?asin=0471317160&revisionId=&format=4&depth=1

[2] Oxtoby, J. C. (2013). Measure and category: A survey of the analogies between topological and measure spaces (Vol. 2). Springer Science & Business Media.

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