Let us denote $D=[0,1]^{n}$ and let $A=\bigg\{(x\_{1},...,x\_{n})\in D^{n}\,,\exists i\,,s.t\,,1<i\leq n\, ,x\_{i}\in\{0,1\}\bigg\}\cup\bigg\{\{1\}\times[0,1]^{n-1}\bigg\}$. Then I want to prove that $D/\sim$ with $A$ identified to a point is homeomorphic to $D$ .

I am having trouble as to produce an explicit homeomorphism(or even a continuous surjection from $D^{n}$ to itself which identifies $A$ with a point and makes $\{0\}\times[0,1]^{n-1}$ the boundary of $D^{n}$ .

In terms of CW complexes , I can say that the space consists of a single $0$-cell with the $n-1$-cell $D^{n-1}$ attached to it by the constant map(So that it is homeomorphic to $S^{n-1}$) . And then we attach the $n$-cell $D^{n}$ with the boundary $S^{n-1}$ being attached to $S^{n-1}$ of the $n-1$ skeleton by the identity map. So it is homeomorphic to $D^{n}\cong D$ .

I tried to do it for the $n=2$ case by mapping $\{0\}\times[0,1]$ to the boundary of $S^{1}$ by $(0,x)\mapsto e^{2i\pi x}$ and $\bigg\{\{1\}\times[0,1]\bigg\}\cup\bigg\{[0,1]\times\{0\}\bigg\}\cup\bigg\{[0,1]\times\{1\}\bigg\}$ to the point $(1,0)$ . Now this map $p:\partial(D)\to D^{2}$ is such that $p\_{\*}$ is the trivial homomorphism so it extends continuously to a map $p:D\to D^{2}$ but how do I tell if it is surjective or anything else?

I have also considered by transforming the square to a hemisphere with boundary(lower) such that the boundary represents . (in the case of D^{2}, I want $[-1,1]$ represent the lower boundary which we will identify with the three edges of the square) . After that I want to draw lines from a point say $p=(0,0,...,0,-2)$ to the (upper) boundary and extend this line to meet $p$ and then use the universal property of the quotient to identify the lower boundary with the point $(0,0,...,0-2)$ to get a bijection which will be a homeomorphism because our space is compact , hausdorff and so is the target.

Here is a picture which will perhaps explain a bit more on what I was trying to do:-
[![enter image description here](https://i.stack.imgur.com/V6w6B.png)](https://i.stack.imgur.com/V6w6B.png)

But as I said, even in this case the explicit map is very complicated and does not allow us to readily view continuity or bijectivity. It's all in terms of lines which depend on the end point. I want to see an explicit map which would help me in constructing these in the future. Most books do not provide explicit maps and say in words however for an initial course in topology and algebraic topology, these things help us to see what's happening more explicitly.

**Why I need this:** I am trying to show that for $n\geq 2$ every cts $f:S^{n}\to X$ extends to a cts map $F:D^{n+1}\to X$ iff $\Pi\_{n}(X,x\_{0})$ is trivial. Where $X$ is a path connected space.

To prove the converse I will need the above result.