Compact Set

Open Cover

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• is a set, is Open Set for every

• Then is known as open cover of .

Example

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• Then is open cover of .

Subcover

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• is open cover of

• Then is subcover of

Finite Subcover

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• If is finite¹ then is finite subcover of

Proof: the open cover of from above example does not have a finite subcover.

Solution

Proof

then is not subcover of

Compact Set

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We say a set  is compact if every open cover of  has a finite sub cover.

Footnotes

1. Finite set has a finite number of members. i.e. ↩