Download University of Bergen General Functional Analysis Problems 5 1) Let

University of Bergen
General Functional Analysis
Problems 5
1) Let B be a base for the topological space (X, τ ). Then x ∈ E if and only if for every B ∈ B with
x ∈ B, there is a y ∈ B ∩ E.
2) Let X satisfy the first axiom of countability (there is a countable base at each point). Then x ∈ E if
and only if there is a sequence from E that converges to x.
3) Let X satisfy the first axiom of countability. Then x is a cluster point of a sequence {xn } from X if
and only if {xn } has a subsequence that coverges to x. Is it true for an arbitrary topological space?
4) Show that f is continuous if and only if it is continouos at each point of a topological space X.
5) Let Bx be a base at x and Cy be a base at y = f (x). Then f is continuous at x if and only if for each
C ∈ Cy there is a B ∈ Bx such that B ⊂ f −1 (C).
6) Prove that if X is a topological space satisfying the second axiom of countability, then any open cover
of X contains a finite or countable subcover of X.
7) Prove that a topological space X satisfies the axiom T1 if and only if every set consisting of a single
point is closed.
8) Prove the Urysohn Metrization Theorem Every normal topological space satisfying the
second axiom of countability is metrizable.
Use the following. Let {Un } be a countable base for the topology. Let (Uni , Umi ) be an enumeration
of all pairs of elements in this base such that U n1 ⊂ Umi . For each i let fi be a continuous function
satisfying 0 ≤ fi ≤ 1 and such that fi is 0 on U n1 and 1 on the complement of Umi . Let
d(x, y) =
∞
X
1
|fi (x) − fi (y)|.
2i
i=1
Show that d is a metric and that the identity mapping is continuous with respect to the given topology
on X and the topology obtained from the metric. Use the fact that given x ∈ X and some open set Um
in the base containing x, there exists another set Un in the base such that
x ∈ Un ∈ U n ⊂ Um .
9) Let A be a connected subset of a topological space, and suppose that A ⊂ B ⊂ A. Then B is
connected.
10) A space X is said to be arcwise connected if given to points x and y in X there is a continuous map
f : [0, 1] → X such that f (0) = x and f (1) = y.
1. Show that an arcwise connected space is connected.
2. In the plane R2 consider the subspace
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X = {(x, y) ∈ R2 : x = 0, y ∈ [−1, 1]} ∪ {(x, y) ∈ R2 : y = sin , x ∈ (0, 1]}.
x
Show that X is connected but not arcwise connected.
3. Show that each connected open set G ∈ Rn is arcwise connected. Use the following argument.
Let c ∈ G and let H be the set of points G that can be connected to x by a polygonal arc. Then
H is open and closed in G.
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