Norms - 8

Question 1

Define a cover and a subcover of a set A

Answer 1

Question 2

When is a cover open?

Answer 2

When all elements are open

Question 3

Define compact

Answer 3

A topological space T is compact if every open cover has a finite subcover

Question 4

State the Heine-Borel theorem

Answer 4

Any closed interval [a,b] is a compact subset of R

Question 5

Lemma: Any closed subset S of a compact space T is compact

Answer 5

Question 6

Lemma: Any compact subset K of a Hausdorff space T is closed

Answer 6

Question 7

Lemma: Any compact subset K of a metric space (X,d) is bounded

Answer 7

Question 8

A subset of R is compact if and only if it’s closed and bounded

Answer 8

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

Question 9

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

Answer 9

Question 10

What is the basis for the product topology T x S, where T and S are both compact topological spaces

Answer 10

Question 11

State Tychanov’s theorem

Answer 11

The product of any collection of compact spaces is compact

Question 12

State and prove Heine-Borel’s theorem in Rⁿ

Answer 12

Question 13

Theorem: A continuous image of a compact space is compact

Answer 13

Question 14

Theorem: A continuous bijection of a compact space T onto a Hausdorff space S is a homeomorphism

Answer 14

Question 15

Define lower semicontinuous and upper semicontinuous

Answer 15

Question 16

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

Answer 16

Question 17

Define a Lebesgue number

Answer 17

Question 18

Define uniformly continuous

Answer 18

Question 19

Theorem: A continuous map from a compact metric space into a metric space is uniformly continuous

Answer 19

Question 20

Define sequential compact

Answer 20

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

Question 21

Lemma: If K is a sequentially compact subset of a metric space then any open cover K has a Lebesgue number

Answer 21

Question 22

If H and K are compact subspaces of a topological space then their union is compact

Answer 22

Let U be an open cover of HuK. Then U is an open cover of H, so theres a finite subcollection U^h that covers H, and a finite subcollection U^k similarly that covers K. Then U^huU^k is a finite subcollection that covers HuK

Question 23

Lemma: The image of a sequentially compact space under a continuous map is sequentially compact

Answer 23