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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.
Table of Contents
Proofs: 1
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References
Bibliography
 Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
 Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition