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Definition: Dense Sets, Nowhere Dense Sets
Let $(X,\mathcal O)$ be a topological space. A subset $U\subset X$ is called:
* dense in $X$ if every point of $X$ is a point or a limit point of $U,$ i.e. if $X$ equals the closure of $U$, formally $X=U^,$
* nowhere dense in $X$ if no nonempty open subset of $X$ is contained in $U^.$
Notes
 From the above and the respective related definitions, it follows immediately that the closure $U^$ of a nowhere dense set $U$ has an empty interior.
 The rational numbers $\mathbb Q$ are dense in real numbers $\mathbb R.$
Mentioned in:
Definitions: 1
Proofs: 2
Propositions: 3
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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