Simmons' Topology and Modern Analysis

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Discussion Overview

The discussion centers on the study of point-set topology and its relationship with modern analysis, specifically through the lens of Simmons' book "Introduction to Topology and Modern Analysis." Participants explore the scope of material to cover in a 16-week semester and share their experiences with various topology texts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks advice on how much material to cover from Simmons' book in a 16-week course, suggesting the first seven chapters on topology as a potential focus.
  • Another participant encourages a leisurely pace and enjoyment of the material, noting that Simmons' book is accessible compared to others like Rudin.
  • A different participant expresses skepticism about the relevance of topology, questioning its practical applications compared to other fields like fractal geometry.
  • One participant expresses enthusiasm for the journey of studying topology, suggesting it will be a rewarding experience.
  • Another participant mentions their preference for Munkres' topology book, indicating uncertainty about its comparative standing.
  • One participant recommends Munkres for clarity and Janich for motivation, while cautioning that Janich lacks definitions for many concepts.

Areas of Agreement / Disagreement

Participants express a mix of enthusiasm and skepticism regarding topology. While some advocate for the enjoyment and exploration of the subject, others question its practical significance. There is no consensus on the best approach to studying topology or the relative merits of different textbooks.

Contextual Notes

Participants' views on the relevance and clarity of topology vary, and there are differing opinions on the appropriate pace and depth of study. Some comments reflect personal preferences for specific textbooks, which may not align with others' experiences.

Who May Find This Useful

This discussion may be useful for students considering an independent study in topology and analysis, as well as those seeking recommendations on textbooks and study approaches in these areas.

mattmns
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I am looking at an independent study next semester and I want to do something with point-set topology and analysis, so I have been looking at the book Introduction to Topology and Modern Analysis by Simmons, which I have heard good things about. For a little background I am currently taking Advanced Calculus and we are using Terrence Tao's Analysis I book, so I know a little analysis, but we are doing everything in the real line, nothing in metric/topological spaces. The question I must ask is how much should I try to cover in a 16-week semester? I am thinking the first 7 chapters [Sets and Functions, Metric Spaces, Topological Spaces, Compactness, Separation, Connectedness, Approximation] (the entire section on Topology) would be plenty (I could probably cover the first chapter before the semester begins since it will be nearly all review). For those who have gone through the book, or know of it, does this seem like a good amount for a 16-week course? Either way, what would you suggest? Thanks.
 
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i suggest having as much fun as possible, and going at an any pace you find suitable. this si not a race, and it is very lengthy. so even if you start slowly, if you enjoy yourself you may well outlast others who start more quickly.

simmons is a good, readable, intro to basic topology used in analysis. i recommend it. i liked it myaelf when i found other books like rudin too dry and repellant. if you plan to be a topologist you might like a book with less dry, more fun topology, like artin and braun.
 
i never understood the point of topology. volume of this=volume of that?

at least it has more applications than fractal geometry
 
geez. get on the train. you have a great adventure ahead of you.
 
I'm really liking Topology by Munkres. Not sure where that ranks though.
 
If you want clearness, i suggest Munkres.
If you want motivation, I suggest Janich, but this book does not define many things. Please be ware.
 

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