How can I prove that two sets are isomorphic, short of finding the isomorphism?

real-analysis general-topology

I have a set $E \subset X$ within a metric space ($X, d$). I want to prove that it is isomorphic to $\mathbb{R}^{n \times n}$, in the sense that there exists a continuous bijection between the two. Because $E$ is a fairly complicated set, it would be a huge pain to actually find an exact bijection, so instead I hope to identify a sufficient suite of conditions that I can test $E$ for that will suffice to show that the two are isomorphic.

Is there some sort of known method for doing this? Or will I have to find the exact function?

Thanks.


Solution 1:

The necessary and sufficient set of conditions for $E$ to be homeomorphic to $\mathbb R^{m}$ (in your situation $m=n^2$):

  • $E$ must be a topological manifold of dimension $m$
  • $E$ must be contractible
  • $E$ must be simply connected at infinity

More details here.

Related

Integral $\int_0^{\pi/2} x\cot(x)dx$, Differntiation wrt parameter only.
Solving the recurrence $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{2}\log(n)$
Volumes using triple integration
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$
If $S$ is a finitely generated graded algebra over $S_0$, $S_{(f)}$ is finitely generated algebra over $S_0$?
Factor $x^5+x^2+1$ into irreducible polynomials in $Z[x]$
Why is tensor from a vector space covariant, not contravariant?
What's the relationship between Borel set and set whose boundary is measure zero?
Random Variable absolute value distribution (PDF and CDF)
Convergence of $ \sum_{n=1}^{\infty} (\frac{n^2+1}{n^2+n+1})^{n^2}$
Evaluate $\lim_{x \rightarrow 0} x^x$
Conjugacy classes and centralizers of a SmallGroup