PROPERTIES OF COMPACT SETS

Compact Set :

A subset K of a metric space X is called compact if every open cover  has a finite sub cover.

The compact sets has come properties .

 Here we see their properties with proofs.

Property 1. Closed Subsets Of Compact Sets are Compact .

proof :

Property 2 : The intersection of any collection of compact subsets of a metric space with finite intersection property is non-empty.

Proof :

 

Property 3: Compact subsets of metric spaces are bounded.

Proof :

 

property 4 : Compact subsets of metric spaces are closed.

Proof :        

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