Lecture Notes Topology (2301631) Phichet Chaoha Department of ...

CHAPTER 4
Compactness
Definition 4.1. Let X be a space and C a collection of subsets of X. We say
that C is a covering of X (or C covers X) if ∪ C = X. A subcollection C that still
covers X is called a subcovering of C. If each element of C is open in X, we will
call C an open covering. Moreover, a space X is said to be compact if every open
covering has a finite subcovering.
Definition 4.2. When X is a subspace of Y , we define a [open] covering of X
in Y to be a collection of [open] subsets of Y whose union contains X. Therefore, a
(open) covering of a space X in the previous definition is simply a [open] covering
of X in X.
Theorem 4.3. Let X and Y be spaces such that X ⊆ Y . Then X is compact
iff every open covering of X in Y has a finite subcovering.
Proof. Suppose X is compact and let C be an open covering of X in Y .
It follows that the collection {A ∩ X | A ∈ C} forms an open covering of X (in
X). Then, by compactness of X, it has a finite subcovering, says {A 1 ∩ X, A 2 ∩
X, . . . , A n ∩ X}. Now, it is clear that {A 1 , A 2 , . . . , A n } is a finite subcovering of C.
Conversely, assume that every open covering of X in Y has a finite subcovering.
Let D be an open covering of X (in X). For each D ∈ D, there is an open subset U D
of Y such that D = U D ∩ X. Then, by axiom of choice, we have an open covering
{U D |D ∈ D} of X in Y and, by assumption, it reduces to a finite subcovering, says
{U D1 , U D2 , . . . , U Dn }. Now, it is easy to verify that {D 1 , D 2 , . . . , D n } is a finite
subcovering of D. Therefore, X is compact.
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Example 4.4. Any finite discrete space is compact, but an infinite discrete
space is not.
Example 4.5. R is not compact because {(n, n+2) | n ∈ Z} is an open covering
of R that does not have a finite subcovering.
Example 4.6. A finite union of compact spaces is compact.
Exercise 4.7. Show that { 1 n | n ∈ N} is compact, but { 1 n
| n ∈ N} is not.
Remark 4.8. There are some other types of compactness : a space X is called
• countably compact if every countable open covering of X has a finite subcovering.
• limit point compact if every infinite subset of X has a limit point.
• sequentially compact if every sequence in X has a convergent subsequence.
It is immediate from the definitions that compactness implies countable compactness.
In general, one can show that (see [4] for details):
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