General exams Say true or false : 1. All topologies on X= {a} are discrete. ( ) 2. No topologies we can construct on the set X= { }. ( ) 3. We cannot define a proper topology on the set of natural numbers N= {1, 2, 3,…}. ( ) 4. Only three topologies we can construct on the set X= {a, b}. ( ) 5. All topologies on a nonempty set X must contain a proper subsets of X. ( ) 6. Let N be the set of all positive integers. Say true or false (a) 1 consists of N, , and every set {1, 2, …., n}, for any positive integer n is a topology on N. ( ) (b) 2 consists of N, , and every set {n, n + 1, …. }, for any positive integer n, topology on N. ( ) is not a 7. All possible topologies on X = {a, b} are 3. ( ) 8. Only three topologies on Y= {a, b, c} are quasidiscrete. ( ) 9. On Y= {a, b, c} there exist two topologies which their union is also a topology on Y. ( ) 10. Let X be an infinite set and a topology on X. If every infinite subset of X is in , then is the discrete topology. ( ) 11. Let R be the set of all real numbers. Then (a) 1 consists of R, , and every interval (a, b), for a and b any real numbers with a < b; is not a topology on R. ( ) (b) 2 consists of R, , and every interval (-r, r), for r any positive real numbers; is a topology on R. ( ) (c) 3 consists of R, , and every interval (-r, r), for r any positive rational numbers; is a quasidiscrete topology on R. ( ) 12. If (X, T) is a discrete space or an indiscrete space then every open set is a clopen set. ( ) 13. Let X be an infinite set. There are no topologies other than the indiscrete topology on X is finite. ( ) 14. Let X be an infinite set and T a topology on X with the property that the only infinite subset of X which is open is X itself. Is (X, T) necessarily an indiscrete space? ( ) 15. The topology T = { X, , { 0 } }on the set X = {0,1} is a 0 -space but not a 1 -space. ( ) 16. Each topology which is a 0 -space must be a 1 -space. ( ) 17. Let X be any infinite set. The countable-closed topology is a 1 -space. ( ) 18. Let 1 and 2 be two topologies on a set X and they are 0 - spaces. If is defined by = ∩ then is not necessarily a 0 -topology on X. 3 3 1 2 3 ( ) 19. If (X, 1 ) and (X, 2 ) are 1 -spaces, then (X, 1 ∩ 2 )is also a 1 -space. ( ) 20. A topological space (X, T) is said to be separable if it has a dense subset which is countable. Then (a) R with the usual topology is separable ( ) (b) A countable set with the discrete topology is separable ( ) (c) A countable set with the finite-closed topology is separable ( ) (d) (X, T) where X is finite is separable ( ) (e) (X, T) where T is finite is separable ( ) (f) An uncountable set with the discrete topology is separable ( ) (g) An uncountable set with the finite-closed topology is separable ( ) (h) A space (X, T) satisfying the second axiom of accountability is separable ( ) 21. Let (X, T) be any topological space and A any subset of X. the largest open set contained A is called the interior of A and is denoted by int (A). Then (a) In R, int ([0, 1]) = (0, 1). ( ) (b) In R, int((3, 4)) = (3, 4). ( ) (c) If A is open in (X, T) then int {a} = {a }. ( ) (d) In R, int ({3}) = . ( ) (e) If (X, T) is an indiscrete space then, for all proper subsets A of X, int (A) = . ( ) (f) For every countable subset A of R, int (A) = . ( ) 22. If A is any subset of a topological space (X, T), the int(A) = X ( X A) . ( ) . Is the property of being a i space i 0,1, 2, 2,3 is a topological property. ( ) is 23. A space x . for every (i) (ii) D O is called a D space G (iii) ) If ) if for every and a closed set false ( if x/ is a closed set Then is true ( a open set D space F x ,there exists an such that x G F is ) 1 D is true ( is compact and ) F is closed, then F is compact. ( 24. Let ) 1 ,......, m be 1 ...... m ( compact subsets of a space . then is also compact . ) 25.Compactness, sequentially compactness countably compactness, weakly countably compactness and locally properties. ( ) compactness are topological