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...
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...
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism. mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
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...
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...
1. Consider the unit circle: (x,y) : x2 y2 = 1. T. Let f R2 ->R be defined by f(x,y) = x2-y, and let F : R2 -> R be defined by F(x, y) Compute the integral of f and F around the unit circle. For the integral of F, proceed in the standard (anticlockwise) direction
DIVERGENCE THEOREM Evaluate using the DIVERGENCE theorem Let S be the denote the portion of the graph of the function z-x2 + y2 between the heights 3 and 5. A parameterisation of this surface is r(u, u) = (u cos u, u sin u, t,-) with u E [0, 2π] and u ε [V3, V5]. Let the orientation of this surface have normal with negative z coordinate. Let F-(y,-r,e) be a vector field over R. Let S be the denote...
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...
-Yi+ 6. Let F = Fii+F23 = 3. 202 + y2 22 + y² (a) Let C denote the unit circle x2 + y2 = 1 oriented counterclockwise. Calculate ScFodř without using Green's Theorem. That is, parameterize C and evaluate this line integral directly.
Let X,Y be topological spaces, and f:X->Y a homeomorphism, I.e. f is one-to-one, onto, and f and f-1 are continuous. a.) Prove that if X is T4, so is Y. b.) Prove that if X is separable, then so is Y.