I am stuck as to how I find the homeomorphism described above. $\delta$ is here described to be the metric $$\delta((m_1,n_1), (m_2,n_2))= max\{d_M(m_1,m_2),d_N(n_1,n_2)\}$$ a metric on $M\times N$ where $(M,d_m)$ and $(M, d_N)$ are metric spaces. I then do not know how to find this homeomorphism, and where really to start?
Any help appreciated, I really struggle with finding homeomorphisms. Thanks.
A homeomorhpism is a continuous bijection with continuous inverse. The obvious candidate for homeomorphism here is the identity $\mathrm{id}\colon[0,1]^2\to [0,1]^2$.
To show continuity of function between metric spaces, we need to show that the preimage of any open ball is open. Identity function is its own inverse, so preimage of any set is that set itself.
Thus, to finish the proof, you need to take any open ball in $([0,1]^2,\delta)$ and show it's open in Euclidean metric, and vice versa: take any open ball with respect to Euclidean metric and show it's open with respect to $\delta$.
Can you finish it from here?
(Hint: Open balls wrt Euclidean metric are standard balls, and open balls wrt to $\delta$ are squares.)