Nowhere dense
In topology, a subset A of a topological space is called nowhere dense if the interior of the closure of A is empty. For example, the integers form a nowhere dense subset of the real line R.Note that the order of operations is important. For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it isn't nowhere dense; in fact it is dense in R, which is essentially the opposite notion.
Note also that the surrounding space matters: a set A may be nowhere dense when considered as a subspace of X but not when considered as a subspace of Y.
Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense. That is, the nowhere dense sets form an ideal of sets[?]. The union of countably many nowhere dense sets, however, need not be nowhere dense. Thus, the nowhere dense sets need not form a σ-ideal[?].
The concept is mainly important to formulate the Baire category theorem.
A substitute shines brightly as a king,
Empties itself, as does an inland brook
CONSCIENCE.
Thus conscience does make cowards of us all;
Is sicklied o'er with the pale cast of thought;
With this regard, their currents turn awry,
CONTENT.
My crown.html">crown is in my heart, not on my head;
Nor to be seen; my crown is called "content";
CONTENTION.
How, in one house,
Hold amity?
King Lear -- II. 4.
Neither supreme, how soon confusion
The one by the other.
Coriolanus -- III. 1.
And range with humble livers in content,
And wear a golden sorrow.
King Henry VIII. -- II. 3.
The valiant never taste of death.html">death but once.
Julius Caesar -- II. 2.
Of habit's devil, is angel yet in this:
He likewise gives a frock, or livery,
And that shall lend a kind of easiness
For use almost can change the stamp of nature,
With wondrous potency.
Hamlet -- III. 4.
More honored in the breach, then the observance.
Idem -- I. 4.
For that's the end of human misery.
King Henry VI., Part 1st -- III. 2.
It seems to me most strange that men should fear;
Will come, when it will come.
Julius Caesar -- II. 2.
Makes us rather bear those ills we have,
The sense of death is most in apprehension.
Measure for Measure -- III. 1.
Will seize the doctor too.
Cymbeline -- V. 5.
An evil soul, producing holy witness,
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