Let's consider a topological space $X$. We define $X \times I / {\sim}$ as a product of $X$ and a $I=[0, 1]$ closed interval's quotient by the following equivalence relation: $(x, 1) \sim (y, 1), (x, 0) \sim (y, 0)$, $\alpha_{1} \neq \{0, 1 \}$, $(x, \alpha_{1}) \sim (y, \alpha_{2}) \leftrightarrow (x=y) \wedge (\alpha_{1}=\alpha_{2})$
Informally speaking, if we consider $X = \mathbb{S}^{1}$ and denote $Y= \mathbb{S}^{1} \times [0, 1]/ {\sim}$ we see that it looks like a two glued cones,which have a common basement.
The problem is to show that $\mathbb{S}^{n} \times I / {\sim}$ is homeomorphic to $\mathbb{S}^{n+1}$.
The trouble is that to cope with it rigorously. The common approach is to try looking for a bijection, which is continous and its inverse is also so, but this doesn't looks as a more or less general method, which enables to solve various problems, requiring homeomorphysm's proof. Also, sometimes it is possible to apply a quotient universal property, which tell us the answer without explicit building of a map.
The questions are: 1) How can we prove that two spaces are homeomophic without building an exact map?
2) How to prove that two spaces are NOT homeomorpic, but thier fundamental groups are isomorphic (without applying other homotopy groups). (sometimes it's possible to notice that one is compact/connected and the second is not but not always)
3) How to find the approach to the exact problem, which is described above?
Any sort of help would be much appreciated.