Textbook recommendations on geometry & topology

[MTB]

Hello fellows

My background is architecture (bachelor in2016) but for unknown reasons I’ve been fascinated by geometry since last year. it was roughly at the stage where I was trying to grasp ‘the truth ‘ of architecture and somehow got into geometry... happy coincidence.

Since I hadn’t touched math after high school and forgot most of the math knowledge anyway... I picked up “Art of problem solving - Introduction to Geometry” as a starter.

Ultimately I’d like to get into topology too.

I wonder if i started off the right foot and use what books to get to the final goal. like right now i don't know which books i should be reading and in what sequence. And what subjects I should be studying( like any linear algebra etc?)

Based on high school and current experience, I am good at geometry but suck at algebra, doing ok with calculus. I am considering math as at least part of my career in the future. also reading the thirteen books of the elements by Sir Thomas L. Heather and How to prove it by Velleman. Any suggestions would be appreciated.

 

[The Bill]

I definitely recommend that you study linear algebra. It is very useful for practical applications of geometry, and studying it will give you a greater intuition about the structure of geometric spaces. Also, linear algebra and concepts learned while studying it are used in some of the basic examples of topological spaces, like metric spaces, vector spaces, and inner product spaces.

For a first introduction to linear algebra, I like to recommend older editions of David C. Lay's text, such as his Linear Algebra and Its Applications, 3rd Updated Edition. That edition covers the basics well, has good explanations, and the newer editions aren't any better. Also, used copies of that edition are very inexpensive.

To help solidify intuition watch the 3Blue1Brown videos on linear algebra on YouTube.

Linear algebra is a pretty good gateway to the way mathematicians think about math, too. After you've learned some of that, you can start learning basic topology without feeling like you got tossed in at the deep end without preparation.

For a basic introduction to topology that actually gets into the real math, I recommend A Combinatorial Introduction to Topology by Michael Henle. It goes slow compared to texts aimed at mathematics students, and it covers the material honestly.

 

[MTB]

I definitely recommend that you study linear algebra. It is very useful for practical applications of geometry, and studying it will give you a greater intuition about the structure of geometric spaces. Also, linear algebra and concepts learned while studying it are used in some of the basic examples of topological spaces, like metric spaces, vector spaces, and inner product spaces.

For a first introduction to linear algebra, I like to recommend older editions of David C. Lay's text, such as his Linear Algebra and Its Applications, 3rd Updated Edition. That edition covers the basics well, has good explanations, and the newer editions aren't any better. Also, used copies of that edition are very inexpensive.

To help solidify intuition watch the 3Blue1Brown videos on linear algebra on YouTube.

Linear algebra is a pretty good gateway to the way mathematicians think about math, too. After you've learned some of that, you can start learning basic topology without feeling like you got tossed in at the deep end without preparation.

For a basic introduction to topology that actually gets into the real math, I recommend A Combinatorial Introduction to Topology by Michael Henle. It goes slow compared to texts aimed at mathematics students, and it covers the material honestly.

Thanks Bill really appreciate ur help. I will definitely look into those books u recommended. Feels there's a long way ahead of me but really excited !