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Definition: Open and Closed Functions
Let $(X,\mathcal O_X)$ and $(Y,\mathcal O_Y)$ be topological spaces. A function $f:X\to Y$ is said to be open closed) if the image $f[A]$ of each open set $A$ in $X$ is open (closed) in $Y.$
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Definitions: 1
Proofs: 2 3
Propositions: 4 5
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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