The following proposition provides a means to construct a topology by hand, just by providing a subbasis.

Proposition: Construction of Topological Spaces Using a Subbasis

Let $X$ be a set and let $S$ be an arbitrary set of its subsets. Then there is exactly one topology $\mathcal O(S)$, for which the set $S$ is a subbasis $\mathcal O(S)$ it is called the topology generated by the subbasis $S.$

Proofs: 1

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