MAT237Y1 Study Guide - Topological Data Analysis, Open Set, Closed Set
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1.2 topology Closure of a set
Closure of a set S, denoted by Sis the union of the set Stogether with boundary points of S:S=S∪∂S.
Interior and closure of a set are in a dynamic relationship, and most of the topological analysis takes place in
this relationship. Of course, interior of a set Sdenoted by Sint is an open set, that is
x∈Sint ⇐⇒ ∃r > 0such that B(r, x)⊂S
This statement characterizes the interior of a set. Whereas the characteristic of a point in the closure of Sis
x∈S⇐⇒ ∀r > 0B(r, x)∩S6=∅
See how the negation of this statement implies that the complement of the closure of a set Sis the
interior of the complement of S:
∃r > 0such that B(r, a
a)∩S=∅(which means B(r, a
This implies (using proposition 1.4,) that the closure of a set is a closed set
An important characterization of the closure of a set is given in 1.14, which claims any point in the closure
of Sis the limit point of a sequence of points in S. Now combine this idea with the fact that S=Sto
conclude that if a sequence of points in Sconverges (to a point a
a) then a
amust be in S, so that there must
be a sequence from Sthat converges to the point a
a. What does this mean?
Some famous examples of closure of a set are:
•closure of any closed set is the set itself
•Let Sbe the set of rational numbers as a subset of R. Then S=R, while Sint =∅. Note also that in
this case Sint =R, while Sint =∅.
•The previous example demonstrates that in general it is not true that for a set S,S=Sint. But if for
some set Swe have S=Sint then such a set will be very special set which is very important for the
theory of integration in chapter 5. A region of plane or of space for which this property holds is known
as a regular region (see page 222.)
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