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Proof

(related to Proposition: Alternative Characterization of Topological Spaces)

• By applying the De Morgan rules to the set $\mathcal C$ we get exactly the definition of a topology based on open subsets of $X.$
• Thus $(X,\mathcal C)$ is a topological space.
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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