Consider the set A = {(x,0) ER|XE R} in R2 with the vertical line topology. Find...
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...
please explain in detail, for each part, especially for parts (g)-(i). *Here is the information about the basis for (h) and (i), it is the basis that generates the vertical interval topology on R^2: -5 JO1 Sewn 2.1 (2.1. Determine Int(A) and Cl(A) in each case. (a) A = (0, 1] in the lower limit topology on R. (b) A = {a} in X = {a,b,c} with topology {X, Ø, {a}, {a, b}}. A = {a,c} in X = {a,b,c}...
on X with 7TCT'. What topology imply about compactness in the Q6. (a) Let X be a set and T,T' two topologies does compactness of X in one other? (b) Show that if X is compact and Hausdorff under both T and T', then either T T', are not comparable they or (c) Consider R with the cofinite topology. Is 0,1 compact? Can you describe the compact sets? (d) Consider R with the cocountable topology. Is 0, 1 compact? Can...
Algebra Consider the following subsets of R2: C1 = {(2, ) ER: x + 2y < 4, x > 0,y 0} C2 = {(x,y) € R2 : 2x + y < 4, x > 0, y 20} Draw a sketch of the intersection CinC, and the union CUC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down.
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...
Let the universal set be R, the set of all real numbers, and let A {xE R I -3 sxs 0, B {xER -1< x 2}, and C xE R | 5<xs 7}. Find each of the following: (a) AUB {xR-3 < x2} s -3orx > 과 xs. (b) AnB xR-12 {*E찌-1 <xs마 frER< -1 orx {*ER|x s -1 or*> 아 (c) A {*ER-3 <x< 아} {*ER|-3 < 아} s-3 orx> 아 frER< 3 orx x s 0 (d) AUC...
(3E) Slinky Line. Consider R with the usual topology. Prove that R/N (see 3.12(e)) is Hausdorff but not first countable. (3E) Slinky Line. Consider R with the usual topology. Prove that R/N (see 3.12(e)) is Hausdorff but not first countable.
A weird vector space. Consider the set R+ = {x ER : x > 0} = V. We define addition by x y = xy, the product of x and y. We use the field F = R, and define multiplication by co x = xº. Prove that (V, O, RO) is a vector space.
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...
Consider the following subsets of R2: C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0} C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0} Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each...