Referring to standard textbooks, one finds frequent appeals to readers’ intuition, when fundamental theorems are proved or basic definitions are explained.
What exactly is intuition in such an abstract environment as topology, and how it can be developed?
The notion of 'intuition' can depend on whether a philosophical or psychological perspective is taken. From a perspective of the philosophy of mathematics, mathematics develops through a particular mathematical intuition shared by all humans but linguistic representations of mathematics have departed from the intuitive mathematical constructions. For more on this see the chapter "Mathematics as intuition" is this recent book:
Ravn, O., & Skovsmose, O. (2019). Connecting Humans to Equations. Cham, CH: Springer
Intuition is the ability to see what is reasonable or not (see https://rjlipton.wordpress.com/2010/10/01/mathematical-intuition-what-is-it/ ). Developing intuition is just like developing any other thinking ability.
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