Question

Show that if Y is G-delta set in X , and if X is a compact Hausdorff space, then Y is a Baire space in subspace topology.

Answer #1

Show that if X is an infinite set, then it is connected in the
finite complement topology.
Show that in the finite complement on R every
subspace is compact.

(a) If G is an open set and K is a compact set with K ⊆ G, show
that there exists a δ > 0 such that {x|dist(x, K) < δ} ⊆
G.
(b) Find an example of an open subset G in a metric space X and
a closed, non compact subset F of G such that there is no δ > 0
with {x|dist(x, F) < δ} ⊆ G

Show that a compact subset of a Hausdorff space is closed in
detail.

Suppose that f : X → Y and g: X → Y are
continuous maps between topological spaces and that Y is Hausdorff.
Show that the set A = {x ∈ X : f(x) = g(x)} is closed in X.

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

1. Consider the set U={(x,y) in R2| -1<x<1 and y=0}. Is U
open in R2? Is it open in R1? Is it open as a subspace of the disk
D={(x,y) in R2 | x^2+y^2<1} ?
2. Is there any subset of the plane in which a single point set
is open in the subspace topology?

Exercise 9.1.11 Consider the set of all vectors in R2,(x, y)
such that x + y ≥ 0. Let the vector space operations be the usual
ones. Is this a vector space? Is it a subspace of R2?
Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy =
0. Is this a subspace of R2? Is it a vector space? The addition and
scalar multiplication are the usual operations.

Suppose G acts on a set X, and x,y ∈ X. Show that either O(x) =
O(y) or O(x) ∩ O(y) = ∅.

Is the set of all x, y, z such x+ 3y + 2z = 0 a subspace of R^3
? If so find a basis for the space.

Let (X,d) be a metric space which contains an infinite countable
set Ewith the property x,y ∈ E ⇒ d(x,y) = 1.
(a) Show E is a closed and bounded subset of X. (b) Show E is
not compact.
(c) Explain why E cannot be a subset of Rn for any n.

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