Question

Show that the following collections τ of subsets of X form a topology in the given space.

a) Let X = R with τ consisting of all subsets B of R such that R\B contains finitely many elements or is all of R.

b) Let X = {a, b, c} and τ = {∅, {c}, {a, c}, {b, c}, {a, b, c}}.

Answer #1

Show that an intersection of finitely many open subsets of X is
open.

Let T be the half-open interval topology for R, defined in
Exercise 4.6.
Show that (R,T) is a T4 - space.
Exercise 4.6
The intersection of two half-open intervals of the form [a,b) is
either empty or a half-open interval. Thus the family of all unions
of half-open intervals together with the empty set is closed under
finite intersections, hence forms a topology, which has the
half-open intervals as a base.

let Xn be a sequence in a metric space X . If Xn -> x in X
iff every neighbourhood of x contains all but finitely many points
of the terms of {Xn}

Let W be the subset of R^R consisting of all functions of the
form x ?→a · cos(x − b), for real numbers a and b. Show that W is a
subspace of R^R and find its dimension.

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Which of the following subsets of R form a field (justify)
{ a + b cube root (2)| a, b ∈ Q }

Let X be a set and let (An)n∈N be a sequence of subsets of X.
Show that: (a) If (An)n∈N is increasing, then liminf An = limsupAn
=S∞ n=1 An. (b) If (An)n∈N is decreasing, then liminf An = limsupAn
=T∞ n=1 An.

Let X be a set and A a σ-algebra of subsets of X.
(a) A function f : X → R is measurable if the set {x ∈ X : f(x)
> λ} belongs to A for every real number λ. Show that this holds
if and only if the set {x ∈ X : f(x) ≥ λ} belongs to A for every λ
∈ R. (b) Let f : X → R be a function.
(i) Show that if...

Let G be the subgroup of R^3 consisting of all vectors of the
form (x, y, 0). Let G act on R^3 by left multiplication. Describe
the orbits of this G-action geometrically. Show that the set of
orbits are in one to one correspondence with R

A)Let the Universal Set, S, have 118 elements. A and B are
subsets of S. Set A contains 18 elements and Set B contains 94
elements. If the total number of elements in either A or B is 95,
how many elements are in B but not in A?
B)A company estimates that 0.3% of their products will fail
after the original warranty period but within 2 years of the
purchase, with a replacement cost of $350.
If they offer...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 6 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago