please do part 4? Page 109, # 37. Page 110, # 38. Let f:(X, taux)rightarrow(Y, t

Question

please do part 4?

Page 109, # 37. Page 110, # 38. Let f:(X, taux)rightarrow(Y, tauy) be continuous, and let be given, We let fA denote the restriction of f to A, i.e. Domain(fA) = A and fA(a) = f(a) Prove the mapping fA:(A, tauA)rightarrow(Y, tauy) is continuous, where tauA denotes the relative topology. If f is also an open mapping, what must be true about A in order for fA to be open? Once you've answered that, do exercise # 40 on page 110. Page 110, # 42 and # 43. (I've already done part(i) of # 42.) Let X = Y = R with the usual topology, and consider f:XrightarrowY defined by f(x) = |x|. Find the subbase sigma for X which is generated from f, i.e. find all possible values of f-1(V) given any open interval V = (a,b), and then list the eight sets in the topology generated by sigma. Page 110, # 44.

Explanation / Answer

claim: f_A is continuos

proof: Let V be open in Y i.e V in au_Y

Then, f^{-1}_(A) (V) = f^{-1} (V) n A , {here n denotes intersectin}

and thus it lies in au_X, the topology of A

Hence, f_A is continuous

Next,

Suppose f is open mapping, then for f_A to be open, it should map an open set to open set,

so, au_A must be subset of au_X, so, A is an element of au_X, or A must be open in X

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