Prove that the open balls (disks) in R^2 form a basis for the standard (product) topology.
1. Prove that no proper subset of RRR is simultaneously open and closed in the RR topology). 2. Prove that no proper subset of RFC is simultaneously open and closed.
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...
Topology Prove that if X and Y are connected topological spaces, then X x Y with the product topology is connected
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2
Prove: If a function is differentiable on an open set U, then it is continuous on U.
Prove that if a topology ? has a countable subbase, then it also has a countable base.
What is the open chain version of the Haworth projection? Convert the Haworth projection for a monosaccharide to its corresponding Fischer projection. CH2OH - 0 OH Select the Fischer projection that is the open chain version of the Haworth projection. oH O
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...