f:X-Y continuous map, kex a compact 70 If sef, then f(K) is compact
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- 18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.Suppose the function f: X --> Y is onto. Prove or disprove that the induced map f-1: P(Y) --> P(X) is onto. As a first step, make sure to state what it means that a function is onto.Show that if a contraction map f: from M to M has a fixed point then it's unique.
- Let f: X->Y and g:Y->X be maps between sets. Suppose that g ∘ f is equal to the identity map id_X: X->X. Does it follow that f ∘ g is equal to the identity map id_Y:Y->Y? Give a proof or a counterexample.1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is continuous if O2 ⊂ O1. (2) Show that the identity map idX : (X, O1) → (X, O2) is not continuous if O1⊂≠O2.Which of the following prescriptions does not define a linear mapping A: R^2 ---> R^2?
- Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the graph {(x,f(x))∣x∈X} is closed.Let ƒ: S→R be an uniformly continuous function. Finish the proof by showing that the limit lim_(x →b)ƒ(x) exists.Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique.
- Prove this is an equivalece relation. x related to y iff x2 - y2 is evenLet X be f(x) = x^(2/3) , X is a subset of R x R satisfying the given equation. Define a bijective map g : X -->R. Show that your map g is well-defined, injective,and surjective.Find an example of a function f : [−1,1] → R such that for A := [0,1], the restrictionf |A(x) → 0 as x → 0, but the limit of f(x) as x → 0 does not exist. Show why