Let R be the real line with the Euclidean topology. (a) Prove that R has a...

Let R be the real line with the Euclidean topology.

(a) Prove that R has a countable base for its topology.

(b) Prove that every open cover of R has a countable subcover.

Expert Solution

The answer is in the pic. If any doubt still remained, let me know in the comment section.

If this solution helped, please don't forget to upvote to encourage us. We need your support. Thanks ☺☺☺ :)

Colby Messinger answered 2 years ago

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and...

TOPOLOGY Let f : X → Y be a function. Prove that f is one-to-one and onto if and only if f[A^c] = (f[A])^c for every subset A of X. (prove both directions)

In RU (R is the reals, U is the usual topology), prove that any open interval...

In RU (R is the reals, U is the usual topology), prove that any open interval (a, b) is homeomorphic to the interval (0, 1). (Hint: construct a function f : (a, b) → (0, 1) for which f(a) = 0 and f(b) = 1. Show that your map is a homeomorphism by showing that it is a continuous bijection with a continuous inverse.)

Prove that the discrete topology on X is the same as the metric topology induced by...

Prove that the discrete topology on X is the same as the metric topology induced by the discrete metric. Where metric topology is defined as: If (X,d) is a metric space, then consider the collection T of all open subsets of X. Then (X,T) is topological space. This topology is called the metric topology on X induced by d.

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be...

Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2) Consider the subset B={−1/n | n∈N, n≥1}⊆A. Determine an upper bound for B in A. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: least upper bound is an upper...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles....

Topology (a) Prove that the interval [0,1] with the subspace topology is connected from basic principles. (b) Prove that the interval [0,1] with the subspace topology is compact from basic principles.

. Let Π be a finite incidence geometry. Prove that, if every line in Π has...

. Let Π be a finite incidence geometry. Prove that, if every line in Π has exactly n points and every point in Π lies on exactly n + 1 lines, then Π is an affine plane. Come up with a similar criterion for finite geometries satisfying (EP) (those geometries are called projective planes).

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if...

Let (an) be a real sequence in the standard metric. Prove that (an) is bounded if and only if every subsequence of (an) has a convergent subsequence.

Question