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.
Let S 1 be the unit circle with the usual topology, S 1×S 1 be the...
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...
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...
Let S2 denote the 2-dimensional sphere. Define the complex projective line 1 as the quotient space 2 \ {0} / ∼ , where ∼ is the equivalence relation on 2 \ {0} that x ∼ y if x = λy for some λ∈C. Prove that S2 and 1 are homeomorphic. Let S denote the 2-dimensional sphere. Define the complex projective line CP as the quotient space C {0}/~, where is the equivalence relation on {0} that I ~y if r...
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 S = {(x, y) = RP.22 + y2 = 1} denote the unit circle in R2 with the subspace topology. Define the function F: (0,1) + S via th (cos(2), sin(24t)) Prove that F is one-to-one, onto, and continuous, but not a homeomorphism.
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...
3. (a) Let (R, τe) be the usual topology on R. Find the limit point set of the following subsets of R (i) A = { n+1 n : n ∈ N} (ii) B = (0, 1] (iii) C = {x : x ∈ (0, 1), x is a rational number (b) Let X denote the indiscrete topology. Find the limit point set A 0 of any subset A of X. (c) Prove that a subset D of X is...
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.
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
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...