Jul 7, 2022 · Explain why this part of mathematics is worth taking now, or working toward, as opposed to classes on extremal combinatorics, representation theory, Fourier analysis, mathematical logic, or machine learning. I think topology is cool …

It’s just bread-and-butter. Everybody needs point-set topology, but after learning the basics you learn whatever flavor of topology is appropriate for your field, e.g. paracompact spaces for differential geometry, Zariski-type spaces for algebraic geometry, banach or hilbert spaces for functional analysis.

May 31, 2024 · Mathematics undergrad while pursuing a Masters in Math for Teaching. In none of the courses I’ve taken have I been “formally” introduced to topology. Can someone explain, briefly, what topology is and perhaps recommend a short textbook to go through on my own time? Thanks in advance. I’m a huge fan of this community.

Haven't read it myself, but people who have seem to love it. I just skimmed through it and it seems fantastic and the choice of topics is very good if you wish to learn topology in search of understanding more geometrical side of mathematics: differential geometry, algebraic topology, algebraic geometry, complex analysis, etc.

Although it focuses on manifolds, but you will learn a great deal (possibly more than Munkres) of point-set topology in the first 5 chapters. After doing this, I don't feel that you need to study point-set topology further unless you want to do research in it. Better move to other, related areas such as algebraic topology, functional analysis, etc.

In a metric space a topology based on open balls can be established, where an open ball is a set of points whose distance from a given point is less than a given value. It can be shown that the intersection of a set of open balls comprised of all open balls with the same center and with radii greater than some fixed value is a closed set.

Some geometric aspects of algebraic topology can be demonstrated with pen/paper/scissors/tape. (Jordan curve) Q: Draw a loop on a piece of paper that doesn't cross over itself (explain continuity as not lifting the pen). Does th

Apr 11, 2012 · I am fascinated by topology because topology is all about starting with one of the most simple and basic mathematical ideas (just open sets and basic structures), we can learn and study the incredibly intricate and complex ways in which they …

Sep 23, 2022 · munkres starts out pretty good, but i feel the quality kind of declines after a while. mendelson is cheap, and is good if you want a small book containing the essentials. kelley is a bit of an older book, but i think it is written really well. lee is new and also written well, but a lot of propositions are left as exercises, most of the book ...

The right topology to consider, it turns out, was the initial topology generated by the linear extensions of all continuous functions from the Cantor space to R. (By way of a quick definition: the initial topology generated by a set of functions is the finest topology that makes all of those functions continuous.