Hi. I'm having trouble with this question in my Topology class.
Can I get some help on this?? Thank you.
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Hi. I'm having trouble with this question in my Topology class. Can I get some help on this?? Tha...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Let X = {(x, y) ∈ R2: x ≥ 0 or y = 0}; and let τ be the subspace topology on X induced by the usu...
Let X = {(x, y) ∈ R2: x ≥ 0 or y = 0}; and let τ be the subspace topology on X induced by the usual topology on R2 . Suppose R has the usual topology and we define f : X → R by f((x, y)) = x for each (x, y) ∈ X. Show that f is a quotient map, but it is neither open nor closed.(So, a restricted function of an open function need not be...
Hi there, I literally got stuck on this question, it would be great if someone can give me help, ...
Hi there, I literally got stuck on this question, it would be
great if someone can give me help, many thanks in advance!
A polynomial on R is a function p : R -R of the form p(x) - aj' where each ai E R and at most finitely many ai are nonzero. Let P denote the set of all polynomials on R (a) What is the dimension of P, regarded as a vector space over R? You do not...
New problems for 2020 1. A topological space is called a T3.space if it is a...
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
For Topology!!! Match the terms and phrases below with their definitions. X and Y represents topological spaces. Note: there are more terms than definitions! Terms: compact, connected, Hausdorff, hom...
For Topology!!!
Match the terms and phrases below with their definitions. X and Y represents topological spaces. Note: there are more terms than definitions! Terms: compact, connected, Hausdorff, homeomorphis, quotient topology, discrete topology, indiscrete topology, open set continuous, closed set, open set, topological property, separation, open cover, finite refinement, B(1,8) 20. A collection of open subsets of X whose union equals X 20. 21. The complement of an open set 21. 22. Distinct points r and y can be separated...
Have to get an idea of how i am doing on this problem. Whould be nice...
Have to get an idea of how i am doing on this problem. Whould be
nice to get a good explaination for each part of the problem. d1
and d2 is the two different metrics, p ,Y.
Problem 2. Consider first the following definition: Definition. Let X be a set and let pand be two metrics on X. We say that p and are equivalent if the open balls in (X, p) and (x,y) are "nested". More precisely, p and...
(TOPOLOGY) Prove the following using the defintion: Exercise 56. Let (M, d) be a metric space...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
The question that is being asked is Question 3 that has a red rectangle around it....
The question that is being asked is Question 3 that has a red
rectangle around it.
The subsection on Question 7 is just for the Hint to part d of
Question 3.
Question 3. Lul (X', d) be a metric space. A subsct ACX is said to be Gy if there exista a collection of open U u ch that A- , , Similarly, a subact BCis said to be F if there exista collection of closed sets {F}x=1 such...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Topology
3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Hi there, I literally got stuck on this question, it would be
great if someone can give me help, many thanks in advance!
A polynomial on R is a function p : R -R of the form p(x) - aj' where each ai E R and at most finitely many ai are nonzero. Let P denote the set of all polynomials on R (a) What is the dimension of P, regarded as a vector space over R? You do not...
New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show...
Consider R with the usual Euclidean topology and let I = [0, 1] be the closed unit interval of R with the subspace topology. Define an equivalence relation on R by r ~y if x, y E I and [x] = {x} if x € R – I, where [æ] denotes the equivalence class of x. Let R/I denote the quotient space of equivalence classes, with the quotient topology. Is R/I Hausdorff? Is so, prove so from the definition of...
For Topology!!!
Match the terms and phrases below with their definitions. X and Y represents topological spaces. Note: there are more terms than definitions! Terms: compact, connected, Hausdorff, homeomorphis, quotient topology, discrete topology, indiscrete topology, open set continuous, closed set, open set, topological property, separation, open cover, finite refinement, B(1,8) 20. A collection of open subsets of X whose union equals X 20. 21. The complement of an open set 21. 22. Distinct points r and y can be separated...
Have to get an idea of how i am doing on this problem. Whould be
nice to get a good explaination for each part of the problem. d1
and d2 is the two different metrics, p ,Y.
Problem 2. Consider first the following definition: Definition. Let X be a set and let pand be two metrics on X. We say that p and are equivalent if the open balls in (X, p) and (x,y) are "nested". More precisely, p and...
(TOPOLOGY) Prove the following using the defintion:
Exercise 56. Let (M, d) be a metric space and let k be a positive real number. We have shown that the function dk defined by dx(x, y) = kd(x,y) is a metric on M. Let Me denote M with metric d and let M denote M with metric dk. 1. Let f: Md+Mk be defined by f(x) = r. Show that f is continuous. 2. Let g: Mx + Md be defined...
The question that is being asked is Question 3 that has a red
rectangle around it.
The subsection on Question 7 is just for the Hint to part d of
Question 3.
Question 3. Lul (X', d) be a metric space. A subsct ACX is said to be Gy if there exista a collection of open U u ch that A- , , Similarly, a subact BCis said to be F if there exista collection of closed sets {F}x=1 such...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
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