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
open.)
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Let X = {(x, y) ∈ R2: x ≥ 0 or y = 0}; and let τ be the subspace topology on X induced by the usu...
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...
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...
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...
Let and define by . Describe the quotient topology on induced by by assuming that has...
Let and define by . Describe the quotient topology on induced by by assuming that has the usual topology.
Problem 2.36. Let X be a space and Y CX. Give Y the subspace topology. Describe...
Problem 2.36. Let X be a space and Y CX. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y.
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous, Let X and Y be top...
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let S1 be the unit circle with the usual topology, S1 × S1 be the product...
Let S1 be the unit
circle with the usual topology, S1 × S1 be
the product space, and define the torus T : = [0,1] × [0,1] / ∼ as
a quotient space, where ∼ is the equivalence relation that (1,y) ∼
(0,y) for all y ∈ [0,1] and (x,0) ∼ (x,1) for all x ∈ [0,1]. Prove
that S1 × S1 and T are homeomorphic.
Let Sl be the unit circle with the usual topology, Stx St be the...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing funct...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
Let S 1 be the unit circle with the usual topology, S 1×S 1 be the...
Let S 1 be the unit circle with the usual topology, S 1×S 1 be
the product space, and define the torus T := [0, 1] × [0, 1]/ ∼ as
a quotient space, where ∼ is the equivalence relation that (1, y) ∼
(0, y) for all y ∈ [0, 1] and (x, 0) ∼ (x, 1) for all x ∈ [0, 1].
Prove that S 1 × S 1 and T are homeomorphic.
(20 points) Let Sl be...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
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...
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...
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...
Problem 2.36. Let X be a space and Y CX. Give Y the subspace topology. Describe (in a useful way) the closed sets in Y.
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Let S1 be the unit
circle with the usual topology, S1 × S1 be
the product space, and define the torus T : = [0,1] × [0,1] / ∼ as
a quotient space, where ∼ is the equivalence relation that (1,y) ∼
(0,y) for all y ∈ [0,1] and (x,0) ∼ (x,1) for all x ∈ [0,1]. Prove
that S1 × S1 and T are homeomorphic.
Let Sl be the unit circle with the usual topology, Stx St be the...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
Let S 1 be the unit circle with the usual topology, S 1×S 1 be
the product space, and define the torus T := [0, 1] × [0, 1]/ ∼ as
a quotient space, where ∼ is the equivalence relation that (1, y) ∼
(0, y) for all y ∈ [0, 1] and (x, 0) ∼ (x, 1) for all x ∈ [0, 1].
Prove that S 1 × S 1 and T are homeomorphic.
(20 points) Let Sl be...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
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