NMT - 9 - 10 Flashcards
Define a partition of a topological space T
Define connected
Give the 4 equivalent statements to T being disconnected
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
When is a subset S of T connected/disconnected
When (S, ts) is connected/disconnected
Define separated
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
If I is a subset of R, when is it an interval
x,y in I and x < z < y implies that z is in I
Theorem: The continuous image of a connected set is connected
Theorem: The product of two connected spaces is connected
Define a path and path connected
Proposition: A path-connected space is connected
Define completeness
A metric space (X,d) is complete is every cauchy sequence in X converges
Proposition: Any compact metric space (X,d) is complete
Theorem: Rd is complete
What is the space Cb(T)
all bounded continuous functions from any topological space T into R
Theorem: Any metric space (X,d) can be isometrically embedded into the complete metric space B(X)
Define a contraction
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
State the contraction mapping 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
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
Define a totally bounded metric space
When is a subspace Y of a metric space totally bounded
Lemma: A subspace of a totally bounded metric space is totally bounded
Lemma: If a subspace Y of a metric space X is totally bounded then so is Yclosure
Proposition: Any sequence in a totally bounded metric space (X,d) has a cauchy sequence
Theorem: A subspace Y of a complete metric space (X,d) is compact if and only if it’s closed and totally bounded
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
Define equicontinuous and uniformly equicontinuous
Lemma: If X is compact the S, a subset of C(X) is equicontinuous if and only if its uniformly equicontinuous
State the Arzela-Ascoli Theorem
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
define diam(S)
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
State and prove Cantors Theorem
Lemma: If (xn) is a Cauchy sequence in a metric space (X,d) has a subsequence that converges to some x in X then xn tends to x
