Spaces

The motivation of this post is to enlarge the steps of theorems and the definition of relativeness, compactness and more. Since the famous and brilliant book Principles of Mathematical Analysis by Walter Rudin is so terse so to say. I love “Baby Rudin”, but unfortunately it is hugely incomprehensible if you do not know what you are doing. Rudin, of course, had deliberately skipped over the details. He probably guessed that the reader was quite gifted in mathematics or the reader is taught by a brilliant teacher. ...

First Initiation When we first started to learn mathematics, our concern is only to solve problems, understand what equation means, comprehend what teacher says during class and etc. At high school we learn more about sets and functions. However, if we are not curious enough, then the philosophical meaning of the word “mathematical space” is not emphasized. When $f: \mathbb{R} \rightarrow \mathbb{Q}$ is written, everybody who has learned high school math can say “the function takes values from $\mathbb{R}$ and maps it to $\mathbb{Q}$. But have people reading this ever thought about the background? ...