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closure∗
mathwizard†
2013-03-21 13:02:37
The closure A of a subset A of a topological space X is the intersection of
all closed sets containing A.
Equivalently, A consists of A together with all limit points of A in X or
equivalently x ∈ A if and only if every neighborhood of x intersects A. Sometimes the notation cl(A) is used.
X
If it is not clear, which topological space is used, one writes A . Note that
X
Y
if Y is a subspace of X, then A may not be the same as A . For example, if
X
Y
X = R, Y = (0, 1) and A = (0, 1), then A = [0, 1] while A = (0, 1).
∗ hClosurei created: h2013-03-21i by: hmathwizardi version: h31191i Privacy setting: h1i
hDefinitioni h54A99i
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