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Unformatted text preview: Math 117 – May 05, 2011 10 Compact sets Deﬁnition 10.1. A family of sets F is called an open cover of a set S , if all the sets in F are open,
and their union contains S . A subset S ⊆ F is called a subcover of S , if it is also an open cover of S .
Using the notion of a cover and a subcover, we can deﬁne compact sets.
Deﬁnition 10.2. A set S is called compact, if every open cover of S contains a ﬁnite subcover.
Although the deﬁnition is abstract and cumbersome to work with, in the space of real numbers R,
there is a simpler characterization of compact sets, given by the HeineBorel theorem below, for the
proof of which one needs the following lemma.
Lemma 10.3. If S ⊂ R is nonempty, closed and bounded, then S has a maximum and a minimum.
Using this lemma one can then establish the following.
Theorem 10.4 (HeinceBorel). A subset S ⊂ R is compact iﬀ S is closed and bounded.
The above characterization of compact subsets of R allows one to establish useful results for special
subsets of R.
Theorem 10.5 (BolzanoWeiestrass). If a bounded set S ⊂ R has inﬁnitely many points, then there
exists an accumulation point of S in R.
The above theorem of course implies that only ﬁnite or unbounded sets may not have accumulation
points in R. Some other useful results are given below.
Theorem 10.6. Let F = {Kα : α ∈ A } be a family of compact subsets of R, such that the intersection
of any ﬁnite subfamily of F is nonempty. Then ∩α∈A Kα = ∅.
And as a simple corollary of this theorem one has.
Corollary 10.7 (Nested intervals). Let F = {An n ∈ N} be a family of closed bounded intervals in R,
such that An+1 ⊆ An for all n ∈ N. Then ∩∞ An = ∅
n=1
. ...
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 Spring '08
 Akhmedov,A
 Sets

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