Topology question: Prove that a bijection f : X → Y is a homeomorphism if and...

Topology question:

Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

Expert Solution

Colby Messinger answered 2 years ago

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)

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove...

Let (X,dX),(Y,dY ) be metric spaces and f: X → Y be a continuous bijection. Prove that if (X, dX ) is compact, then f is a homeomorphism. (Hint: it might be convenient to use that a function is continuous if and only if the inverse image of every open set is open, if and only if the inverse image of every closed set is closed).

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is...

Given f(x,y) = 2 ; 0< x ≤ y < 1 a. Prove that f(x,y) is a joint pdf. b. Find the correlation coefficient of X and Y.

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: X→Y be a map with A1, A2⊂X and B1,B2⊂Y (A) Prove f(A1∪A2)=f(A1)∪f(A2). (B) Prove...

Let f: X→Y be a map with A1, A2⊂X and B1,B2⊂Y (A) Prove f(A1∪A2)=f(A1)∪f(A2). (B) Prove f(A1∩A2)⊂f(A1)∩f(A2). Give an example in which equality fails. (C) Prove f−1(B1∪B2)=f−1(B1)∪f−1(B2), where f−1(B)={x∈X: f(x)∈B}. (D) Prove f−1(B1∩B2)=f−1(B1)∩f−1(B2). (E) Prove f−1(Y∖B1)=X∖f−1(B1). (Abstract Algebra)

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2,...

Let F be a field. (a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) = 2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple. That is, a^2 + b^2 = c^2. Use this fact to explain how to generate right triangles with integer side lengths. (b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy − y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be...

Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one to one and onto functions. Then g ◦ f : X → Z is one to one and onto; and (g ◦ f)−1 = f−1 ◦ g−1 ).

Prove the following. Let T denote the integers divisible by three. Find a bijection f :...

Prove the following. Let T denote the integers divisible by three. Find a bijection f : Z→T (Z denotes all integers).

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function...

Let f: X→Y and g: Y→Z be both onto. Prove that g◦f is an onto function Let f: X→Y and g: Y→Z be both onto. Prove that f◦g is an onto function Let f: X→Y and g: Y→Z be both one to one. Prove that g◦f is an one to one function Let f: X→Y and g: Y→Z be both one to one. Prove that f◦g is an one to one function

Prove or disprove if B is a proper subset of A and there is a bijection...

Prove or disprove if B is a proper subset of A and there is a bijection from A to B then A is infinite

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