The invariance of domain theorem states that if is a one-to-one and continuous map between -manifolds without boundary, then is an open map.
Domain Invariance Theorem
See also
Dimension Invariance TheoremExplore with Wolfram|Alpha
References
Rusin, D. "RE: Homeomorphisms of the Sphere." In The Mathematical Atlas. http://www.math.niu.edu/~rusin/known-math/99/invar_domain2.Referenced on Wolfram|Alpha
Domain Invariance TheoremCite this as:
Weisstein, Eric W. "Domain Invariance Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DomainInvarianceTheorem.html