For each of the following statements, indicate whether it is true or false. Let
ID: 1943923 • Letter: F
Question
For each of the following statements, indicate whether it is true or false. Let X be the set if real numbers that have decimal expansions that do not contain the digit 3. Then X is a closed subset of R (with respect to the usual topology). Let X be the indiscrete topology on it set X and let f : X rightarrow Y be a continuous function (with respect to the indiscrete topology). Then f must be a constant function. Suppose that f : X rightarrow Y is a continuous function of topological spaces mid that G is an open subset of X. Then f(G) must be an open subset of Y. (Recall that f(G) is the image of G under the function f.) A B = A B. int(A B) = int(A) int(B). (int(A)) = int(A). Suppose that f : X rightarrow Y is a continuous function of topological spaces and that A C X. Then f(A) C f(A).Explanation / Answer
1)The statement is FALSE. Clearly 3 is not an element of X. Now you consider the sequence of numbers 2.9,2.99,2.999,2.9999,.... then 3 never comes in the decimal expansion of these numbers, hence this is a sequence in X and clearly they converge to 3 but 3 is not in X. Hence X doesn't contain a limit point implies X can't be closed. 2)No, it may not be a constant function. For example, let X be the space with indiscrete topology and you consider the identitiy function f from X to X i.e. f(x)=x for all x in X. If X has more than 1 element then this function is non constant and also continuous. Because you take x in X then a open set containing x is whole X and f-1(X)=X which is open. Hence since inverse image of open sets under f are open then f is continuous. 4)AnB+its limit points=Closure(AnB) A+its limit pointsnB+its limit points=Closure(A)nClosure(B) If x is in Closure(AnB) and x is limit pt of AnB then .ie. x is limit pt of either A and B(as AnB is subset of A and B) .x is in Closure(A) and Closure(B).Hence x is in Closure(A)nClosure(B).Thus Closure(AnB) is in Closure(A)nClosure(B) If x is in Closure(A)nClosure(B)and x is limit pt then its a limit pt of both A and B.Hence its a limit [pt of AnB. Thus Closure(A)nClosure(B) is in Closure(AnB).Hence Closure(A)nClosure(B)=Closure(AnB). 3)If f(G) is not open then (f^-1)(f(G)) is not open ie G is not open contradiction.Hence true 6)A=(0,1) Closure(A)=[0,1] int(Closure(A))is not equal to [0,1] as it is a closed set. Hence false
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