# nowhere dense set

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## nowhere dense set

[′nō‚wer ′dens ′set]
(mathematics)
A set in a topological space whose closure has empty interior. Also known as rare set.
References in periodicals archive ?
Therefore, [mathematical expression not reproducible] is a fuzzy soft nowhere dense set in (U, E, [psi]).
Also [L.sub.M] [??] [Fr.sup.fs] ([V.sub.D]) is a fuzzy soft nowhere dense set. Therefore, [mathematical expression not reproducible] is a fuzzy soft nowhere dense set as well.
The idea of intuitionistic fuzzy nowhere dense set in intuitionistic fuzzy topological space presented and studied by by Dhavaseelan and et al.
[11] If A is a neutrosophic nowhere dense set in (X, T), then [bar.A] is a neutrosophic dense set in (X, T).
It is known that for an almost continuous function f with a dense graph, every nowhere dense set is f-negligible with respect to AC (see [13]).
A Fuzzy topological space (X, T) is called a Fuzzy nodec space if every non-zero Fuzzy nowhere dense set is Fuzzy closed in (X, T).
Let [[lambda].sub.i] be a Fuzzy nowhere dense set in a Fuzzy nodec space (X, T).
If A [member of] [NCT.sup.[alpha]] we have A = (NC int(NCcl (NC int(A)) [intersection] [(NC int(NCcl(NC int(A)) [intersection] [A.sup.C]).sup.C], where (NCint(NCcl(NCint(A)) [intersection] [A.sup.C]) clearly is neutrosophic crisp nowhere dense set, we easily see that B [subset] NCcl (NC int(A)) and consequently A [subset] B [subset] NC int(NCcl (NC int(A)) so the proof is complete.
For a neutrosophic crisp NC[alpha]-topology may be characterized as neutrosophic crisp topology where the difference between neutrosophic crisp open and neutrosophic crisp nowhere dense set is again a neutrosophic crisp open, and this evidently is equivalent to the condition stated.
To point out the difference between (q)-porous and nowhere dense sets, note that if E [subset] Y is nowhere dense, y [member of]Y and r > 0, then there are a point z [member of] Y and a number s > 0 such that B(z,s) [subset] B(y,r) E.
In fact, if lattice DP(X) is isomorphic to lattice DP(Y) then we obtain a bijective map F: X [right arrow] Y preserving closed nowhere dense sets, which turns out to be a homeomorphism if X and Y are countably compact T3 spaces without isolated points.
We show here that for a Hausdorff space X without isolated points there is a bijection from [LAMBDA] onto X which maps [LAMBDA]-closed sets in [LAMBDA] to closed nowhere dense sets in X.

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