5) Compactness

Question 1

What is a covering

Answer 1

Question 2

What is a subcovering

Answer 2

Question 3

What does it mean to be compact

Answer 3

A subset A ⊆ X is compact if every open coveringof A contains a finite subcovering

Question 4

What is the relationship between finite and compact

Answer 4

If A ⊆ X is finite, then it is compact.

Question 5

What is the relationship between compact and closed

Answer 5

If a subspace A ⊆ X is compact, then it is closed

Question 6

What is the relationship between compact, closed and bounded

Answer 6

. If a subspace A ⊆ X is compact, then it is closed and bounded

Question 7

If we have the closed subspace of a compact metric what do we know about the subspace

Answer 7

The subspace is compact

Question 8

If f : X → Y is continuous, and A ⊆ X is compact what do we know about the image

Answer 8

The image f(A) ⊆ Y is also compact

Question 9

If f : X → R is continuous, and A ⊆ X is nonempty and compact what do we know about f

Answer 9

f is bounded and attains it’s bound. That is, there exist a, b ∈ A such that f(a) ≤ f(x) ≤ f(b) for all x ∈ A

Question 10

What do we know about the union of compact subspaces

Answer 10

If A1, . . . , Ar are compact subsets of X, so is A := A1 ∪ · · · ∪ Ar.

Question 11

Given compact subspaces A1, . . . Aq of a metric space X, determine whether or not A := A1 ∩ · · · ∩ Aq is compact

Answer 11

Since Aj is compact for 1 ≤ j ≤ q, it is closed. Thus A is closed. So A is a closed subset of any of the compact spaces Aj , and is therefore compact

Question 12

What is implied if f : (X, dX) → (Y, dY ) and A is compact in (X, dX)

Answer 12

Question 13

What does it mean to be sequentially compact

Answer 13

Question 14

Given sequentially compact subspaces A1, . . . Aq of a metric space X, determine whether or not A = A1 ∩ · · · ∩ Aq is sequentially compact

Answer 14

Question 15

How do we show that something is not sequentially compact

Answer 15

In each subspace, we need to find an infinite sequence with no convergent subsequences (in that subspace!)

Question 16

Describe the condition under which an infinite sequence in a metric space contains a subsequence that converges to a specific point

Answer 16

Let (xn)n≥1 be an infinite sequence in X, and let
x ∈ X; if, for any ε > 0, the ball Bε(x) contains xn for infinitely many values of n, then (xn)n≥1 contains a subsequence that converges to x in X

Question 17

What is the relationship between compact and sequentially compact

Answer 17

If a subspace A ⊆ X is compact, then it is sequentially compact

Question 18

What does it mean for a metric space to be complete

Answer 18

A metric space (X, d) is complete if every Cauchy sequence converges to a limit in X

Question 19

With regards to which metric is the space C[a,b] complete

Answer 19

The space C[a, b] is complete with respect to the dsup metric

Question 20

What is the relationship between a Cauchy sequence in a metric space and the convergence of its subsequences

Answer 20

Question 21

What do we know about the closed subspace a complete metric space

Answer 21

A closed subspace Y of a complete metric space X is itself complete

Question 22

What is the relationship between compactness and completeness

Answer 22

Any compact metric space X is complete

Question 23

What is the relationship between completeness and closedness

Answer 23

Any complete subspace Y ⊆ X is closed in X

Question 24

In the Euclidean line, (R, d) if A ⊆ R is closed and bounded what do we know about it

Answer 24

Then it is compact

Question 25

In the Euclidean Space (R^n, d2), under what condition is a set A ⊂ R^n compact

Answer 25

If and only if it is closed and bounded