topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly...

interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. Given a topological space...

Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it...

studied a combination of set theory and analysis topics to arrive at the Baire category theorem and the definition of a nowhere dense set. He then used these...

X} will be a topological space. The definition of meagre set uses the notion of a nowhere dense subset of X , {\displaystyle X,} that is, a subset of X...

{\displaystyle \operatorname {J} (f)} is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers)...

holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set. There are...

countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal....

manifold Tensor density in differential geometry Dense set and nowhere dense set Dense-in-itself is a set that contains no isolated points Density (graph...

two dense-in-itself sets is not dense-in-itself. But the intersection of a dense-in-itself set and an open set is dense-in-itself. Nowhere dense set Glossary...

Tychonoff. Nowhere dense A nowhere dense set is a set whose closure has empty interior. Open cover An open cover is a cover consisting of open sets. Open ball...

of nowhere dense subsets, meager subsets, and Baire spaces. A set is the boundary of some open set if and only if it is closed and nowhere dense. The...

discussed a nowhere-dense set of positive measure on the real line, an early version of the Cantor set, now known as the Smith–Volterra–Cantor set. Smith,...

category 1.  A set of first category is the same as a meager set: a set that is the union of a countable number of nowhere-dense sets, and a set of second...

union of compact convex sets is again compact and convex. Meager, nowhere dense, and Baire A disk in a TVS is not nowhere dense if and only if its closure...

perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped...

projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense...

Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty. Any open subspace of a Baire space is itself a Baire space...

Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere dense...

According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension...

small set. In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples...

R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is nowhere continuous, there is a dense subset D {\displaystyle D} of R {\displaystyle \mathbb {R}...

countable union of porous subsets of X. Any porous set is nowhere dense. Hence, all σ-porous sets are meagre sets (or of the first category). If X is a finite-dimensional...

set Cofinal (mathematics) Cofinite Dense set IP set 2-large Large set (Ramsey theory) Meagre set Measure zero Natural density Negligible set Nowhere dense...

Every dense Gδ set in a Baire space is a Baire space. The result need not hold if the Gδ set is not dense. See the Examples section. Every comeagre set in...

Since the set on which a holomorphic function vanishes is closed and has empty interior (by the Identity theorem), a thin set is nowhere dense, and the...

category in Y {\displaystyle Y} (a set is of the first category or meagre if it is the countable union of nowhere-dense sets). If Y {\displaystyle Y} is a...

convex functions, it is a meager set, that is, a countable union of nowhere dense sets. The boundary of any convex polygon forms a convex curve (one that...

property Universally Baire set Meager set Comeager set - A comeager set is one whose complement is meager. Null set Conull set Dense set Nowhere dense set...

in X {\displaystyle X} is nowhere dense, and X {\displaystyle X} is meagre in itself.) In particular, this proves that the set of all real numbers is uncountable...