Explicit homeomorphism from $I \times I$ to itself that maps $I\times \{0\} \cup \{0,1\} \times I$ to $I \times \{0\}$

Question

My question is basic and I apologize in advance if it's too easy.

I've been reading some books on basic topology and I keep seeing the same thing: that there is an "obvious" homeomorphism: $$\varphi: I \times I \to I \times I, $$ that takes $I \times \{0\} \cup \{0,1\} \times I$ to $I \times \{0\}$. But I cannot find an explicit formula.

The statement is quite general and I have found this question but the author says that it's obvious without explaining why. I'm sure it's obvious but I really want to see it.

Can anyone help ?

Thank you.

Answer 1

Here is an illustration of a piecewise linear homeomorphism with the desired property defined on 3 pieces: two triangles and a trapezoid.

On each piece you can obtain an explicit formula by putting any three vertices and their images into the general form of an affine linear map.