NMT - 9 - 10

Question 1

Define a partition of a topological space T

Answer 1

Question 2

Define connected

Answer 2

Question 3

Give the 4 equivalent statements to T being disconnected

Answer 3

i) T has a partition of two non-empty open sets
ii) T has a partition of two non-empty closed sets
iii) T has a subset that is both open and closed and neither T nor the empty set
iv) theres a continuous function from T onto the two point set {0,1} with the discrete topology

Question 4

When is a subset S of T connected/disconnected

Answer 4

When (S, t_s) is connected/disconnected

Question 5

Define separated

Answer 5

Question 6

Proposition: A subspace S of a topological space T is disconnected if and only if it’s separated by some open subsets U,V in t

Answer 6

Question 7

If I is a subset of R, when is it an interval

Answer 7

x,y in I and x < z < y implies that z is in I

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Theorem: The continuous image of a connected set is connected

Answer 11

Question 12

Theorem: The product of two connected spaces is connected

Answer 12

Question 13

Define a path and path connected

Answer 13

Question 14

Proposition: A path-connected space is connected

Answer 14

Question 15

Define completeness

Answer 15

A metric space (X,d) is complete is every cauchy sequence in X converges

Question 16

Proposition: Any compact metric space (X,d) is complete

Answer 16

Question 17

Theorem: R^d is complete

Answer 17

Question 18

Answer 18

Question 19

What is the space C_b(T)

Answer 19

all bounded continuous functions from any topological space T into R

Question 20

Theorem: Any metric space (X,d) can be isometrically embedded into the complete metric space B(X)

Answer 20

Question 21

Define a contraction

Answer 21

A map f: X to X is a contraction if d(f(x), f(y)) <= k d(x,y) x,y in X for some k < 1

Question 22

State the contraction mapping theorem

Answer 22

Let (X,d) be a complete metric space and f: X to X a contraction. Then f has a unique fixed point in X ie there exists a unique x in X such that f(x) = x

Question 23

Theorem: Let (X,d) be a complete metric space and f: X to X a contraction. Then f has a unique fixed point in X ie there exists a unique x in X such that f(x) = x

Answer 23

Question 24

Define a totally bounded metric space

Answer 24

Question 25

When is a subspace Y of a metric space totally bounded

Answer 25

Question 26

Lemma: A subspace of a totally bounded metric space is totally bounded

Answer 26

Question 27

Lemma: If a subspace Y of a metric space X is totally bounded then so is Y^closure

Answer 27

Question 28

Proposition: Any sequence in a totally bounded metric space (X,d) has a cauchy sequence

Answer 28

Question 29

Theorem: A subspace Y of a complete metric space (X,d) is compact if and only if it’s closed and totally bounded

Answer 29

If Y is compact then Y is closed and totally bounded since the open cover { B(x,e) ; x in Y} has a finite subcover

If Y is totally bounded then any sequence in Y has a cauchy subsequence. Since X is complete this subsequence converges; since Y is closed the limit of the sequence lies in Y. So Y is sequentially compact and therefore Y is compact

Question 30

Define equicontinuous and uniformly equicontinuous

Answer 30

Question 31

Lemma: If X is compact the S, a subset of C(X) is equicontinuous if and only if its uniformly equicontinuous

Answer 31

Question 32

State the Arzela-Ascoli Theorem

Answer 32

Let X be a compact metric space. A subset A of C(X) is compact if and only if it’s closed, bounded and equicontinuous

Question 33

define diam(S)

Answer 33

If S is a non-empty subset of a metric space (X,d) then diam(S) = sup d(x,y) over x,y in S

Question 34

State and prove Cantors Theorem

Answer 34

Question 35

Lemma: If (x_n) is a Cauchy sequence in a metric space (X,d) has a subsequence that converges to some x in X then x_n tends to x

Answer 35