Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty...
Example: Let x, y ∈ Rn, where n ∈ N. The line segment
joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R
n
. A subset A of Rn, where n ∈ N, is called convex if it contains
the line segment
joining any two of its points. It is easy to check that any convex
set is
path-connected.
(a) Let f : X → Y be an onto continuous function. If X is
path-connected, prove that Y is path-connected.
(b) Conclude that path-connectedness is a topological
property.
(c) Prove that any path-connected space is connected.
(d) Let G = { ( x,sin (1/x) : 0 < x ≤ 1 }.
Let L = { (0, y) : −1 ≤ y ≤ 1 } Let X = G ∪ L as a subspace of R2
with its usual topology.
Prove that X is connected but not path-connected
Homework Answers
Add Answer to:
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty...
6 6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If...
6
6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If T1 is the topology induced orn Y by di and T2 is the subspace topology on Y (induced by T on X), prove that Ti -T2. [This shows that every subspace...
L maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f b...
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous.
l maps is a quotient map. 4, Let ( X,T ) be a topological...
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...
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...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x,...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d...
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by
Let a continuously...
A. Let (X, d) be a metric space so that for every E X and every...
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Given u 0 in Rn, let L-Spanu). For each y in Rh, the reflection of y...
Given u 0 in Rn, let L-Spanu). For each y in Rh, the reflection of y in L is the point reflyy defined by reflLy 2 projy-y The figure shows that reflyy is the sum of proy andý -y Show that the mapping y- ref y is a linear transformation L = Span{u refly y The refiection of y in a line through the origin Let Ty)- refy2 proy-y. How can it be shown that T(y) is a linear transformation?...
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E ...
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn...
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
6
6. Let (X, d) be a metric space and T the topology induced on X by d. Let Y be a subset of X and di the metric on Y obtained by restricting d; that is, di(a, b) d(a, b) for all a and b in Y. If T1 is the topology induced orn Y by di and T2 is the subspace topology on Y (induced by T on X), prove that Ti -T2. [This shows that every subspace...
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous.
l maps is a quotient map. 4, Let ( X,T ) be a topological...
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...
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...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
Let a continuously differentiable function f: Rn → R and a point x E Rn be given. For d E Rn we define Prove the following statements: (i) If f is convex and gd has a local minimum at t-0 for every d E R", then x is a minimiser of f. (ii) In general, the statement in (i) does not hold without assuming f to be convex. Hint: For) consider the function f: R2-»R given by
Let a continuously...
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Given u 0 in Rn, let L-Spanu). For each y in Rh, the reflection of y in L is the point reflyy defined by reflLy 2 projy-y The figure shows that reflyy is the sum of proy andý -y Show that the mapping y- ref y is a linear transformation L = Span{u refly y The refiection of y in a line through the origin Let Ty)- refy2 proy-y. How can it be shown that T(y) is a linear transformation?...
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E N. If gcd(rn, n) = 1, show that φ s an injective map.
Question 1 [22 marks] (Chapt ers 2, 3, 4, 5, and 6) Let A e Rn be an (n x n) matrix and be R. Consider the problem 1 (P2) min2+ s.t. xe R" 1Ax-bil2 1 where & > O is fixed and Il IIl denot es the 2-norm. Call g.(x)=l|2 the objective function of problem (P2) 1Ax-bl2 i) [3 marks] Compute the gradient of g, and use it to show that the solution xi of this problem verifies (I+EATA)(x)...
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