The words in the title may not be accurate, so I put them in quotes.

Recently I begin reading Hatcher's book on Algebraic topology. I have not studied any "topology" other than basic point set topology. I have much more experience with analysis-related topics (analysis of Banach and Hilbert spaces, for example).

I feel a lot of discomfort when reading sections of geometrical descriptions about how one space is homeomorphic to another. For example, the following paragraph taken from section 1.3 (Van Kampen's Theorem) is causing me some troubles. $K$ is a torus knot, embedded in $\mathbb R^3$. Please ignore $X$; it is not relevant to the discussion below. 
[![enter image description here](https://i.stack.imgur.com/v8TBz.png)](https://i.stack.imgur.com/v8TBz.png)

It is very easy to understand why $\partial D^4 = \partial D^2\times D^2\cup D^2\times \partial D^2$. This is just an exercise of point-set topology. The part after "geometrically" is, however, difficult for me. Although it is quite natural to regard $\partial D^2\times D^2$ as a solid torus, I cannot see any obvious reason why it can be identified with the solid torus bounded by the torus knot $K$. It is also very difficult to see why the complement of this solid torus is homeomorphic to $D^2\times \partial D^2$. (It might be not as difficult as I think, but from time to time, I get stuck on some "obvious" things.) **Whenever I come across such descriptions, I struggle a lot, thinking about how to prove those statements.**

Also, the sets $D$, $S^3$ and so on in the book are not specifically defined (for example, as $D^2=\{(x,y):x^2+y^2\le 1\}$). Although defining $D^2$ specifically is clearly unnecessary, sometimes, the vague concept of $D^2$ causes me some difficulties. **I find it really hard to tell whether the author is making a rigorous statement or a non-rigorous description.**

In the end, in this book, I find it much easier to read proofs than to read remarks and other descriptive sections, due to the problems outlined above. Interestingly, when I read analysis books, the opposite is true: remarks are much easier to read than proofs. **I am wondering why this is the case.**

**How can I overcome those problems/difficulties when reading this book?**

This is not seeking personal advice - I believe that this might be the feeling of many other people as well. Please ask me to clarify if anything is unclear.