Norms - 8 Flashcards
Define a cover and a subcover of a set A
When is a cover open?
When all elements are open
Define compact
A topological space T is compact if every open cover has a finite subcover
State the Heine-Borel theorem
Any closed interval [a,b] is a compact subset of R
Lemma: Any closed subset S of a compact space T is compact
Lemma: Any compact subset K of a Hausdorff space T is closed
Lemma: Any compact subset K of a metric space (X,d) is bounded
A subset of R is compact if and only if it’s closed and bounded
Since R is a metric space any compact subset is bounded, and since it’s Hausdorff any compact subset is closed.
If K is a bounded subset of R then K is a subset of [-R,R] for some R > 0. Then K is a closed subset of the compact set [-R,R] so is compact
Lemma: Let F be a collection of non-empty closed subsets of a compact space T such that every finite subcollection of F has a non-empty intersection. Then the intersection of all sets from F is non-empty
What is the basis for the product topology T x S, where T and S are both compact topological spaces
State Tychanov’s theorem
The product of any collection of compact spaces is compact
State and prove Heine-Borel’s theorem in Rn
Theorem: A continuous image of a compact space is compact
Theorem: A continuous bijection of a compact space T onto a Hausdorff space S is a homeomorphism
Define lower semicontinuous and upper semicontinuous
Theorem: If T is compact and f:T to R is lower semicontinuous then f is bounded below and attains a minimum. If f is upper semicontinous then it’s bounded above and attains a maximum
Define a Lebesgue number
Define uniformly continuous
Theorem: A continuous map from a compact metric space into a metric space is uniformly continuous
Define sequential compact
A subset K of a metric space (X,d) is sequentially compact if every sequence in K has a convergent subsequence whose limit lies in K
Lemma: If K is a sequentially compact subset of a metric space then any open cover K has a Lebesgue number
If H and K are compact subspaces of a topological space then their union is compact
Let U be an open cover of HuK. Then U is an open cover of H, so theres a finite subcollection Uh that covers H, and a finite subcollection Uk similarly that covers K. Then UhuUk is a finite subcollection that covers HuK
Lemma: The image of a sequentially compact space under a continuous map is sequentially compact
