Vtyjkyuk

I saw on wikipedia that quotient topology doesn't behaves well in the sense that $X$ can be metrizable, haussdorf... but $X/_!sim$ not. We can see that gluing $0$ and $1$ in the segment $[0,1]$ gives the circle. But do I really have to prove it or such construction is clear by visualizing ? Same if I take $[0,1]times [0,1]$ and I glue all point of the boundary in one, we "see" that it's going to be the sphere $mathbb S^2$. Do I have to prove it rigorously, or it's not really necessary ?

I'm asking this because in a course notes on manifold I'm reading, the teacher say always : "we see that gluing $A$ and $B$ gives torus, or sphere or any geometric figure", but he doesn't prove it rigorously. I sent him an email yesterday, and he sais that such argument is enough to identify things. Is it really the case ? I mean, if the quotient doesn't behaves well, we have to prove rigorously that such identification gives the sphere or the torus (or anything else), no ?

Can someone gives me an example where we expect that a quotient $X/_sim$ we'll be a specific figure, but in fact it will not be ?

I think, perhaps, you are missing the point of why you would or would not prove that the gluing construction gives the desired shape. We intuitively have an idea of what the gluing operation looks like: if I give you an interval, and tell you to glue the two end points, then you know that the outcome should be the circle. If I give you a square and tell you to identify two opposite sides in opposite directions, you know the outcome should be the Möbius strip.

However, the "gluing operation" is an abstract mathematical operation, applied on an object (a topology) which is often counterintuitive.

Thus, if the construction of the circle or the Möbius strip fails, it is not because we have badly predicted what the outcome shape should be, but because the gluing operation is wrong. The point of proving a few examples early on of the effect of the gluing operation -- at one point I proved in an introductory topology class that identifying the end points of an interval really gives a circle -- is to convince oneself that the gluing operation is really what it promises to be. After doing a few of these, you notice that the construction goes the same way every time, and you trust that it will go the same way with every other pictorial example.

Being "obvious" is always a subjective evaluation, similarly as being "trivial". It depends on your knowledge and your experience. For a beginner less things will be obvious than for a person having been active in mathematics for many years.

Saying that something is obvious can have various reasons, for example

(1) It is a known result which has explicitly been proved, but the proof is very simple so that virtually everybody can work it out.

(2) It is very similar to something you know to be true, and you know that the proven methods apply. This is professional experience.

Of course there is also a danger in taking something for obvious: The devil may be in the details, and professional experience may fail. I am sure that the literature contains numerous examples of erroneous proofs (or even erroneous results) rooting in a feeling of obviousness.

Let us come back from philosophy to your examples concerning "glueing". This is definitively category (1). Other examples are statements like "a circle is homeomorphic to a square", "a teacup is homeomorphic to solid torus", etc.

As Mees de Vries said in his answer, if you explicitly work out some of such examples, you will develop an understanding for what is going on and end with the judgement "Yes, it is obvious".