METRIC SPACES 5 TEST QUESTIONS
AND ANSWERS
What is a covering of s set A?
What is a subcovering? - ANSWER 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? - ANSWER If every Ui that forms a covering
is open.
State the definition of a compact subset. - ANSWER 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. - ANSWER (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... - ANSWER it is compact.
What theorem links compactness and boundedness? - ANSWER If A is
compact then it is bounded.
i.e. compact implies bounded
What links compactness and open/closed sets? - ANSWER Is a subspace is
compact, then it is closed.
If a subspace is compact then.. - ANSWER 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? - ANSWER No. There exists metric spaces with closed and
bounded subsets which are NOT compact.
AND ANSWERS
What is a covering of s set A?
What is a subcovering? - ANSWER 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? - ANSWER If every Ui that forms a covering
is open.
State the definition of a compact subset. - ANSWER 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. - ANSWER (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... - ANSWER it is compact.
What theorem links compactness and boundedness? - ANSWER If A is
compact then it is bounded.
i.e. compact implies bounded
What links compactness and open/closed sets? - ANSWER Is a subspace is
compact, then it is closed.
If a subspace is compact then.. - ANSWER 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? - ANSWER No. There exists metric spaces with closed and
bounded subsets which are NOT compact.
