Problem 2.36. Let X be a space and Y CX. Give Y the subspace topology. Describe...
Problem 1: Let X be a linear space. Let Y CX be a linear subspace. (a) Prove that the map : X+X/Y given by 7(x) = (2) is a linear map. (b) Prove without using the dimension formula or rank-nullity that N = Y. (c) Prove without using the dimension formula or rank-nullity that RX = X/Y.
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...
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 II. i) Let Tı and T2 be two topologies on the same space X. Suppose that T2 is finer than η. If (X,n) is compact, does it follow that (X,2) is compact? Conversely, if (X, T2) is compact, does it follow that (X, Ti) is compact? la. ii) Let Y C X be equipped with the subspace topology. Show that Y is compact in the subspace topology if and only if any cover of Y with open sets in...
For metric spaces and topology Problem II. a) Show that f: X →Y is continuous if and only if f-'(C) CX is closed for every closed C CY b) Then show that a function f: X + Y is continuous if and only if f(A) < f(A) for all ACX
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...
(1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated (1) Let (X,d) be a metric space and A, B CX be closed. Prove that A\B and B\A are separated
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...
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
1. Let (X, d) be a metric space, and U, V, W CX subsets of X. (a) (i) Define what it means for U to be open. (ii) Define what it means for V to be closed. (iii) Define what it means for W to be compact. (b) Prove that in a metric space a compact subset is closed.