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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 non-empty 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

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