Topology 1. Consider the topology on the space X-a, b,c,dy given by Is (X, T) connected?...
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...
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 = {0}U[2,3], and give X the topology Tx = {0,{0}, [2, 3], X}. (a) (10 points) Is X To? Briefly justify your answer. (b) (10 points) Is X Hausdorff? Briefly justify your answer. (c) (10 points) Is X Tz? Briefly justify your answer. (d) (10 points) Is D = ({0} Ⓡ {0}) U ([2, 3] x [2, 3]) a closed subset of X x X with the product topology? Briefly justify your answer.
For Topology!!! Match the terms and phrases below with their definitions. X and Y represents topological spaces. Note: there are more terms than definitions! Terms: compact, connected, Hausdorff, homeomorphis, quotient topology, discrete topology, indiscrete topology, open set continuous, closed set, open set, topological property, separation, open cover, finite refinement, B(1,8) 20. A collection of open subsets of X whose union equals X 20. 21. The complement of an open set 21. 22. Distinct points r and y can be separated...
Questions Answers 10. Problem: (Topology of R2) (a) A is open Consider (b) A is closed A = (-1/n, 1/n) x -1, 1] n=1 (c) B is open OO B = (-1/n, 1/n) x [-1,1] n=1 Which statement is true? (d) B is closed. (a) f((0,1]) is compact 11. Problem: (Continuity) Consider the real valued function (b) f(I0, 1)) is compact xsin(x) x 0 f(x) Questions Answers 10. Problem: (Topology of R2) (a) A is open Consider (b) A is...
topology Note: Symbols have their usual meanings. 1. Show that every indiscrete topological space is locally connected. 2. Give an example of locally connected topological space which is not connected. 3. Show that the intersection of any collection of closed compact subsets of a topological space is closed and compact. (2)
Topology C O, 1 and be the supremum norm (a) Prove that (X || |) is a Banach space. You can assume that (X, | |) is a normed vector space (over R) |f|0supE0.1 \5(x)|.| 4. Let X C (b) Show that || |o0 that the parallelogram identity fails.] on X is not induced by any inner product. Hint: Check for all E[0, 1]. Show that {gn}n>1 (0, 1] BI= {gE X |9||<1} is a compact (c) For every 2...
(a) Suppose K is a compact subset of a metric space (X, d) and x є X but x K Show that there exist disjoint, open subsets of Gi and G2 of (X, d) such that r E Gi and KG2. (Hint: Use the version of compactness we called "having a compact topology." You will also need the Hausdorff property.) b) Now suppose that Ki and K2 are two compact, disjoint subsets of a metric space (X, d). Use (a)...
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.
A topological space X has the Hausdorff property if cach pair of distinct points can be topologically scparated: If x, y E X and y, there exist two disjoint open sets U and U, with E U and y E U and UnU = Ø. (a) Show that each singleton set z} in a Hausdorff space is closed A function from N to a space X is a sequence n > xj in X. A sequence in a topological space...