Proposition: Equivalent Notions of Homeomorphisms

Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be topological spaces and let $f:X\to Y$ be bijective. The following definitions of a homeomorphisms: 1. $f$ is a homeomorphism, i.e. $f$ and its inverse function $f^{-1}$ are both continuous. 1. $f$ is both, continuous and an open function. 1. $f$ is both, continuous and an open function.

Proofs: 1


Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs


References

Bibliography

  1. Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
  2. Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition