Compactness

Definitions:

Open Cover

Collection of open sets, whose union contains the current set

Subcover

Subcollection of the cover that is still the cover

Compact set

A topological space X is called compact if every open cover of X has a finite subcover

Bounded set

If there is a ball containing the set

Motivation

• Dealing with the infinity. Compact set is the next best thing after finite set
• Seeks to generalize the notion of a closed and bounded subset of Euclidean space
• Compactness is intrinsic property of the set

Theorems

1

Finite set is compact

2

Compact set is bounded and closed

3

For $K\subset Y\subset X$, $K$ is compact in $Y\iff K$ is compact in $X$

4

Closed subset of the compact set is compact

Heine -Borel

Bounded and closed sets in ${\mathbb{R}}^n$ are compact

5

K compact $\iff$ every infinite subset has a limit point in K

Cantor

For the collection K of the compact subsets of X (some metric space). If every finite sub collection has a non empty intersection then the intersection of the entire collection is not empty

Proof: By contradiction using the compliment of K - open sets. No intersection⇒ compliment covers the whole space

6

K compact $\iff$ every collection of closed sets that has the Finite intersection property (FIP) has non empty intersection

7

Compactness is equivalent to sequential Compactness

8

Compact metric space is Complete metric space

9

Continuous image of the compact is compact