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Concept
Nowhere dense set
Summary
In mathematics, a subset of a 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 clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense.
A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the , which is used in the proof of several fundamental results of functional analysis.
Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density:
A subset of a topological space is said to be dense in another set if the intersection is a dense subset of is or in if is not dense in any nonempty open subset of
Expanding out the negation of density, it is equivalent to require that each nonempty open set contains a nonempty open subset disjoint from It suffices to check either condition on a base for the topology on In particular, density nowhere in is often described as being dense in no open interval.
The second definition above is equivalent to requiring that the closure, cannot contain any nonempty open set. This is the same as saying that the interior of the closure of is empty; that is, Alternatively, the complement of the closure must be a dense subset of in other words, the exterior of is dense in
The notion of nowhere dense set is always relative to a given surrounding space. Suppose where has the subspace topology induced from The set may be nowhere dense in but not nowhere dense in Notably, a set is always dense in its own subspace topology. So if is nonempty, it will not be nowhere dense as a subset of itself. However the following results hold:
If is nowhere dense in then is nowhere dense in
If is open in , then is nowhere dense in if and only if is nowhere dense in
If is dense in , then is nowhere dense in if and only if is nowhere dense in
A set is nowhere dense if and only if its closure is.
Official source
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