2. Let L={x,y,z}with the topology r={L,ø,{x},{x,y},{z},{x,z}}. Verify whether L is normal or not. Explain your answer in details.
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- LetX={(x, y)∈R×R:xy= 1}.Suppose thatR×Rhas thestandard topology, andXhas the relative topology. Prove that thereare disjoint open subsetsVandWofXsuch thatX=V∪W.Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)Is T ={∅, X, {1, a}, {1, b}, {2, c}, {1, a, b}, {1}}a topology on X = {1, 2, a, b, c}?Why?
- Let X={a, b, c, d, e} and let T={X, non zero, {a}, {a, d}, {a, e}, then (X, T) is neither Hausdorff nor connected?2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined byρ(x, y) = d(x, y)1 + d(x, y)for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and thatρ(x, y) ≤ d(x, y) for all x, y ∈ S.A. Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?
- Show that f(x,y) = xy (a) satisfies a Lipschitz condition on any rectangle a ≤ x ≤ b and c ≤ y ≤ d; (b) satisfies a Lipschitz condition on any strip a ≤ x ≤ b and −∞ < y < ∞; (c) does not satisfy a Lipschitz condition on the entire plane.Let R be bounded by the hexagon with vertices at the points (5,0), (6,1), (6,5), (5,6), (4,5), (4,1). Use symmetries in the figure to determine the center of gravity, (x, y ) .A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?