A subset 
of a topological space 
is compact
if for every open cover of 
, there exists a finite subcover of 
.
즉, S의 open cover들에 대해서, S의 finite subcover(element가 유한한 open cover of S의 subset)가 존재하는 경우, S가 compact하다고 한다.
cover
A family 
of nonempty subsets of 
whose union contains the given set 
(and which contains no duplicated subsets) is called a cover (or covering) of 
For example of cover
there is only a single cover of 
,
namely 
.
However, there are five covers of 
,
namely 
, 
, 
, 
, and 
.
minimal cover
A minimal cover is a cover for which removal of one member destroys the covering property.
Open cover
Example of open cover
- an open cover of the real line, with respect to the Euclidean topology, is the set of all open intervals 
, where 
.
, where .
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