Why is homeomorphism understood as stretching and bending? There are homeomorphisms that cannot be considered as stretching and bending (don't know what the proof of this should look like, as we haven't defined stretching and bending formally, this is my major concern here) - see the bottom of page 22 in A guide of topology   by Krantz.
Even if we accept that some homeomorphisms cannot be understood as stretching and bending (whatever stretching and bending means), we assume that all stretching and bending that can be represented by homeomorphism - why is that? Has anyone proved it?
 A: I’d say just leave it as an informal, but powerful way to intuitively reason about topology.
In an analogous situation, computability and decidability of problems can be defined mathematically, but we never can prove mathematically that these definitions really grasp our intuitive notion of what can be computed. The Church–Turing thesis is a meta-mathematical thesis which says that those definitions really do grasp our intuitive notion of computability.
If viewed as a philosophical statement about the nature of our universe, there really is strong evidence in favor of the Church–Turing thesis: For one, all attempts are formalizing computability have been shown to be equivalent. And on the other hand, as one develops a sense for what is Turing–computable, one finds that anything that looks intuitively computable also looks Turing-computable and vice versa: The intuition for the mathematical concept of computability starts to overlap with the pre-mathematical intuition of computability.
And this is the point of my answer: I think with topology, it’s much the same – only that the thesis in question doesn’t have a name (and is known to be slightly false). If you just start doing topology, you will most probably find that your intuition for bending and stretching and your intuitions for homeomorphisms start to overlap a lot. I think this is the best justification for the cited non-mathematical description of topology as a field.

And now, to address your concerns regarding this:
And as you said, the thesis doesn’t work out completely and I share your concerns about using it to prove stuff. Luckily, I found that most of the stuff I have seen proved using visual bending–stretching arguments can really be proven rigorously using real mathematics. This really helped me a lot to accept the attitude. Maybe this will help you as well.
So my advice to you is to not lose to much sleep over it: Don’t think of it as a definition of the field, but rather as a description and don’t expect stretching and bending to be defined formally.
You may either regard topology as mathematics which formalizes reasoning about things like bending and stretching, or you may regard things like bending and stretching as intuitive notions which help you reason about topology.
A: Here are some comments directed the bountied version with the prompt

Could someone please explain where the idea of stretching and bending relates to the ideas in topology? How did people start thinking of this intuitive in this first place.

This is clearly giving a more historical bend to the question, which seems to have had more of a philosophical flavor. The new formulation of the question is still too philosophical it seems to me (for better or worse); and there are already many nice answers addressing these, so I'll focus on the historical context of two (plus one) matters:

*

*"homeomorphism"

*"stretching and bending"

*That the surfaces obtained from two rectangles by gluing two opposing sides of each rectangle with $n$ and $m$ half-twists are homeomorphic iff $n$ and $m$ are both odd simultaneously (in which case they are both homeomorphic to the Mobius band), or even simultaneously (in which case they are both homeomorphic to the cylinder). This is the example that is referred to in the OP, and it sets up the issue: we have two surfaces both homeomorphic to the Mobius band, but it seems one can't stretch and bend one to get the other one.

Instead of building up to definitive answers to the questions above I'll only point to some sources.

First off, "homeomorphism" seems to have been used first in crystallography; its use in mathematics seems to be unanimously attributed to Poincaré. Here is the OED entry:

Indeed, to triangulate further see https://mathshistory.st-andrews.ac.uk/Miller/mathword/h/ , or more properly Moore's paper The evolution of the concept of homeomorphism" (http://dx.doi.org/10.1016/j.hm.2006.07.006). Of course Dieudonné's A History Of Algebraic And Differential Topology, 1900-1960 is also very relevant to the matter at hand.
Thus the first use and definition of a "homeomorphism" is to be located in the Analysis Situs (p.22-23 of https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf) (by which in fact a "$C^1$ diffeomorphism" is meant).
According to Moore's paper, Poincaré himself changed what he meant by a homeomorphism over time, and without acknowledging that he was doing so.

Again according to Moore's paper, the idea that looking of invariants of "stretching and bending" was already familiar to mathematicians before Poincaré, e.g. Möbius (for this he cites Pont's La Topologie algébrique des origines à Poincaré; Dieudonné also points to this work as the definitive source for topology-before-Poincaré)

Despite the fact that Poincaré was the one who defined a homeomorphism (as a diffeomorphism), item 3. in my list above should have been addressed by the time of Poincaré. Indeed, according to the digital copy of a translation of Analysis Situs (p.2), von Dyck had already classified non-orientable surfaces in 1988, despite the fact that in fact von Dyck was using  a continuous bijection as an equivalence, according to Moore's paper (the orientable ones were classified earlier Möbius). I am extrapolating that the "parity of the half-twists" issue came up (von Dyck's paper is available online to verify or debunk this).

Finally let me mention the "parity of the half-twists" issue has been discussed multiple times here on M.SE, see e.g. (some of these are marginally related)

*

*Topologically distinguishing Mobius Strips based on the number of half-twists ,

*How to define the "full-twist" Möbius strip ,

*Topologically distinguishing Mobius Strips based on the number of half-twists ,

*In how many dimensions is the full-twisted "Mobius" band isotopic to the cylinder? ,

*Distinguishing the Cylinder from a "full-twist" Möbius strip ,

*How is a doubly twisted cylinder different from a cylinder? ,

*Twists and half-twists on ribbons in $\mathbb{R}^4$

On a personal note, I remember being puzzled by this matter when I first started learning topology. I also remember looking at some of the related M.SE discussions above. Still, it didn't make sense to me that the way a circle is the same as a square was to be considered the same to the way a Mobius strip with one half-twist is the same as a Mobius strip with three half-twists. As some of the discussions show above that indeed, eventually there is a difference (existence/nonexistence of an ambient isotopy) which is topological (i.e. it's a difference that can be articulated using only the language of topological spaces and continuous maps), but this difference is not as simple as the existence or nonexistence of a homeomorphism between.
It seems the point inherent to all this is this:

*

*Different mathematicians (or same mathematician at different times) use different phrases and nomenclature to signify certain intuitive ideas (mistakenly at times). Often these intuitive ideas don't exactly work in tandem with each other, nor with the formal definitions. Despite this, it's still good to be aware, and knowledgeable about the sedimentation, if not to appreciate it.

A: One should be a little wary of thinking too much about stretching and bending because in the real world these are continuous processes (there are lots of steps partway through the process) whereas in a homeomorphism there needn’t be steps in between. For example consider two interlocking (but not touching) rings (circles) in Euclidean 3-space. This subspace is homeomorphic to two separate rings. However you can’t stretch and bend the rings apart. Thus we see that the notion of stretching and bending is a bit wishy-washy and doesn’t work quite like topology. It’s also useless in some weird topologies like the cofinite topology. However that doesn’t mean it’s not useful. Just look at lots of examples and allow the idea to guide your intuition. Two spaces are essentially equal if they are homeomorphic so another way of thinking about a homeomorphism is that it matches up points in one space to points in the other, and does so continuously so patches in one space correspond to patches in the other, and the way patches connect to each other in one space is the same as the way they connect to each other in the other. But I don’t think this is so helpful.
Another notion of stretching and bending in topology is called a homotopy. It’s a way of saying that two maps from $X$ to $Y$ can be continuously transformed into one another. It allows one to define the notion of homotopy equivalence which is bit like stretching and bending but more permissive than homeomorphism.
A: Morris Kline introduces in Mathematical Thoughts from Ancient to Modern Times  in chapter 50 The Beginnings of Topology right at the beginning the term homeomorphism as follows:


*

*[Morris Kline] A number of developments of the nineteenth century crystallized in a new branch of geometry, now called topology but long known as analysis situs. To put it loosely for the moment, topology is concerned with those properties of geometric figures that remain invariant when the figures are bent, stretched, shrunk, or deformed in any way that does not create new points or fuse existing points. The transformation presupposes, in other words, that there is a one-to-one correspondence between the points of the original figure and the points of the transformed figure, and that the transformation carries nearby points into nearby points. This latter property is called continuity, and the requirement is that the transformation and its inverse both be continuous. Such a transformation is called a homeomorphism.


This informal characterisation indicates that the mathematical notion of continuity is used to assure nearness of near points of geometrical objects. So, stretching and bending is mathematically addressed by continuity. On the other hand, the mathematical notion of bijection of a mapping is used to address deformations which do not create new points or fuse existing points.
The term homeomorphism is a mathematical notion to describe the transformations of stretching, bending, etc. of geometrical objects in a convenient way. In Intuitive Combinatorial Topology by V.G. Boltyanskiĭ and V.A. Efremovich we have a nice comparison:


*

*[Bolt., Efrem.] It is instructive to compare the notion of homeomorphism and congruence of figures. In geometry we study mappings that preserve distances between points. These mappings are called (rigid) motions. A motion transfers a figure from place to place without changing the distances between its points. Two figures that can be made to coincide by means of motions are said to be congruent. They are regarded as copies of each other, i.e., as being the same from a geometric standpoint. In topology we study more general mappings, namely homeomorphisms. Two homeomorphic figures are regarded as copies of each other, i.e., as being the same from a topological standpoint. Properties of figures unchanged by homeomorphisms are called topological properties, or topological invariants. Topology investigates topological properties of figures.


We observe that topological invariants lie at the heart of homeomorphisms. These are the indicators which describe the essence of homeomorphisms according to our current mathematical knowledge. It might be expected that more and other exciting topological invariants will be found in future developments.
As you already noted, stretching and bending a geometrical object is not necessarily the same as a homeomorphic transformation of it. In both books we are informed that


*

*[Kline] if one takes a long rectangular strip of paper and joins the two short ends, he obtains a cylindrical band. If instead one end is twisted through 360° and then the short ends are joined, the new figure is topologically equivalent to the old one. But it is not possible to transform the three-dimensional space into itself topologically and carry the first figure into the second one.


*[Bolt., Efrem.] It would be incorrect to think that it is possible to bring any two homeomorphic figures in space into coincidence by bending stretching and moving (without cutting and glueing together). ... In order to bring them into coincidence by permissible modifications we must first cut the figure, twist it, and glue together the points that were originally together. This procedure (cutting and appropriate regluing after stretching and after modifying the position of parts of a figure) is often used in topology to demonstrate the homeomorphism of two figures.
The sameness of the disposition of two figures in space (or in a figure that contains them) is made precise by the concept of isotopy. We say of two figures $A$ and $B$ that they are isotopic in a figure $P$ that contains them (or topologically equally disposed in $P$) if there is a homeomorphism of $P$ that takes $A$ to $B$.

The cylindrical band above and the twisted band through  360°  are homeomorphic but not isotopic in space.
In order to prove that two figures are not homeomorphic one uses topological invariants.
A: The methodology of mathematics begins with examine some concept by means of presenting formalizations which satisfy the conflicting constraints of (1) faithfully representing the intuition behind the concept and (2) being amenable to mathematical analysis. We judge our formalizations both by these two criteria and by the richness of the pure mathematics that develops out of them.
Topology arises historically as an attempt to understand the notion of continuity, as developed primarily in 19th century analysis. By Cauchy's era continuous real value functions had been defined as we define them today in terms of epsilons and deltas.  From here one gets metric space, and their abstraction to topological spaces. The intuition of homeomorphism as stretching and bending comes really from an even more special chintzy than metric spaces, namely manifolds. In this context there's little room for doubt that stretching and bendings can be represented by homeomorphisms-it's absolutely part of our intuition that these map nearby points to nearby points.
It seems to me the richer question is of whether general homeomorphisms have anything to do with stretching and bending. The answer is surely no, so the notion of topology has to defend itself on other grounds. In fact there's not a complete consensus that topological spaces are the right abstraction of continuity-various smaller and later classes of spaces have been proposed. But most mathematicians don't seem much concerned with finding the best possible abstraction of a concept. Topology successfully models the theories of manifolds, algebraic varieties, function spaces, and other important topics, so people learn and use it. Perhaps this attitude is too pragmatic for you, in which case there are plenty of alternatives to investigate, from convergence spaces to toposes.
A: I study topology that semester and these questions came to me too. If X,Y are topological spaces and f:X→Y continuous then it can be shown easily that the graph of the function, Gf={(x,f(x):x∈X} with the product topology, is homeomorphic with the domain of f, X. If you see that for functions from real numbers to real numbers (with the ordinary topology) you get a geometric result which is very ok with the intuition about streching. The familiar graphs of functions (of 1 variable) from calculus are homeomorhpic with the real number line. If you ''strech'' the curve of the graph you get a line. Similarly, for function from R^2 to R , you get the image of streching a wrinkled sheet to get a smooth surface. You can also see, why the curve of the circle is not homeomorhpic with the real number line, because if you stretch the circle from two points of it you will get a knot somewhere,(not sure about the last).
