In a general topological space, compactness is defined using open covers: a topological space X is compact if and only if every open cover of X has a finite subcover. This means any collection of open sets that covers X can be reduced to a finite number of open sets that still cover X.
Conditions: X is a topological space.; An open cover is a collection of open sets whose union contains X.; A subcover is a subcollection of the open cover that still covers X.
In a general topological space, compactness is defined using open covers: a topological space X is compact if and only if every open cover of X has a finite subcover. This means any collection of open sets that covers X can be reduced to a finite number of open sets that still cover X.
Conditions: X is a topological space.; An open cover is a collection of open sets whose union contains X.; A subcover is a subcollection of the open cover that still covers X.
Removing the endpoint 0 makes (0,1] noncompact because the infinite subset {1/n} still lies inside (0,1], but its limit point 0 is no longer included in the set. This violates the limit-point compactness condition that every infinite subset must have a limit point within the set.
Conditions: E is the half-open interval (0,1].; S is the infinite subset {1/n : n=1,2,3,...}.; Compactness is judged by the limit-point definition.
Removing the endpoint 0 makes (0,1] noncompact because the infinite subset {1/n} still lies inside (0,1], but its limit point 0 is no longer included in the set. This violates the limit-point compactness condition that every infinite subset must have a limit point within the set.
Conditions: E is the half-open interval (0,1].; S is the infinite subset {1/n : n=1,2,3,...}.; Compactness is judged by the limit-point definition.