Metric Spaces 5 UPDATED ACTUAL
Exam Questions and CORRECT Answers
What is a covering of s set A?
What is a subcovering? - A covering of A is a collection of sets {Ui: i element of I} (finite or
infinite) for which A is a subset of the union of all Ui.
A subcovering is a subcollection {Ui: i element of J} for some J subset I, which also covers A.
What is an open covering of A? - If every Ui that forms a covering is open.
State the definition of a compact subset. - A subset of X, A, is compact if every open covering of
A contains a finite subcovering.
Give two examples of sets that are immediately not compact. - (0,1) is not compact in the
Euclidean line (in fact every open interval is not compact).
The real numbers are not compact/
If A is finite then... - it is compact.
What theorem links compactness and boundedness? - If A is compact then it is bounded.
i.e. compact implies bounded
What links compactness and open/closed sets? - Is a subspace is compact, then it is closed.
If a subspace is compact then.. - it is closed and bounded.
i.e. compact implies closed and bounded
Does the converse of the theorem that says compact imp[lies closed and bounded hold? - No.
There exists metric spaces with closed and bounded subsets which are NOT compact.
Exam Questions and CORRECT Answers
What is a covering of s set A?
What is a subcovering? - A covering of A is a collection of sets {Ui: i element of I} (finite or
infinite) for which A is a subset of the union of all Ui.
A subcovering is a subcollection {Ui: i element of J} for some J subset I, which also covers A.
What is an open covering of A? - If every Ui that forms a covering is open.
State the definition of a compact subset. - A subset of X, A, is compact if every open covering of
A contains a finite subcovering.
Give two examples of sets that are immediately not compact. - (0,1) is not compact in the
Euclidean line (in fact every open interval is not compact).
The real numbers are not compact/
If A is finite then... - it is compact.
What theorem links compactness and boundedness? - If A is compact then it is bounded.
i.e. compact implies bounded
What links compactness and open/closed sets? - Is a subspace is compact, then it is closed.
If a subspace is compact then.. - it is closed and bounded.
i.e. compact implies closed and bounded
Does the converse of the theorem that says compact imp[lies closed and bounded hold? - No.
There exists metric spaces with closed and bounded subsets which are NOT compact.
